They might well have had a balanced force including navy, but you specifically built your military to win the naval game. Now your overspecialized military is having trouble against someone trying to counter it.
On a water heavy map, I think we can agree that a powerful navy is a must if we want to win domination victory for the first couple of eras. Until Atomic Era, we have basically two naval units, one melee and one ranged. There are no other options until Carriers and Subs are unlocked. To gain naval superiority, we build a bunch of these to achieve that. The melee ships eventually upgrade into Destroyers while ranged ships eventually upgrade to Battleships. Due to the wars, they will no doubt gain a lot of experience so it's difficult to replace them. Why is it considered overspecializing when we are stuck with these two options for 6 eras? Why are we penalized for keeping our naval units alive to get lots of experience and be extremely powerful?
They exist, in the form of forests, hills and deserts.
If every tile within three tiles of a city is Forest, Hills and Deserts, then I'd be very annoyed as well as this will greatly slow down my advances. This tend to be the case early game and I have no problem with it. However, these are heavily random with the terrain and where cities might be located due to the resource locations. If this happens because of various factors that somehow made the city difficult to take, I accept it as being a part of the game. More often than not, there's a combination of flat and rough terrain (Forests are practically cut down at this point) so there are certain approaches to launching an assault. I'm not sure when was the last time you played against Russia in late game but those Ostrog slow you down so much. I can accept one AI having that ability. Everyone has it in the form of Mine Field is way too much.
You are rewarded. Your military, civilians and trade units can traverse the ocean unmolested, and your navy can provide support to any invasion as necessary, and help siege cities. This is what navies have always been for.
On higher difficulties, the AIs tend to have a lot of land units on their continent. To disembark land units is usually suicide unless you cleared most of the enemy land forces with your Battleships. Let's not forget that, with the recent change to the Cruisers' range, Battleship is the first ranged naval unit with 2 range. That extra range basically means they should provide some hefty support to weaken enemy land forces so your embarked units can launch an amphibious assault with some degree of success. This support has been made much tedious with the Mine Field. With naval superiority, the idea is that it's a matter of time before you whittle down the enemy forces.
The current problem is that none of these other techs matter because a fleet of Battleships can take any coastal city they want with absolutely no recourse except to have a stronger navy yourself. ~6 Battleships per fleet can take most coastal cities without taking damage.
There's one thing we have to realize. We don't necessarily build a fleet of Battleships from scratch. We have Cruisers that upgrade into them so it's much faster to transition into a fleet of Battleships than build a new navy from scratch. Do we want to give the impression that all the Cruisers we built are now less useful because players have to build a more diverse navy? Are we expected to disband some of our Cruisers/Battleships because they no longer project the same dominance as Cruisers in the eras before?
Every Mine Field is 1 less Cruiser that can shoot back, and 8 less GPT. Those add up pretty significantly once you have a couple coastal cities. For example, if you built this in just 5 cities that is 5 less Cruisers and 40 GPT, enough to probably be able to make the difference between your fleet winning or losing and even needing the mines in the first place. And the building literally says "Enemy Naval Units and Embarked Units must expend 1 extra movement per turn if they move into a tile worked by this city."; everything else with similar language does require a tile to be worked, and if this isn't the case the wording is flat out wrong.
The wording is wrong as I've noticed in my game but that's a minor point at the moment. You are right that each Mine Field is 1 less Cruiser/Battleships you must face. However, I can assure you that, even using your example, those 5 Cruisers/Battleships aren't going to make a difference for the AI against my superior navy. We have seen that the AI doesn't always make the best decisions when it comes to their ranged naval units. As a human, I never build Mine Fields. For AIs, whether or not they build Mine Fields will change little in our conflicts if I'm determined to achieve naval superiority. We'd like to think the AI does war quite well but, regarding navy, they are still lagging behind what their armies are capable of.
Essentially, Mine Fields take a very common degenerate situation and help push it away from being completely abused. And even though they exist I still finish most of my warmonger games with a massive fleet of Battleships taking every coastal city my happiness can handle. If my navy is truly strong enough to warrant me taking coastal cities with ease, Mine Fields don't change much.
That's my point exactly. Mine Field doesn't change the situation if a player's navy is truly strong enough which, for most people, it will be at this point. People don't wait until Atomic Era to get Battleships. They usually begin earlier if necessary but, at the latest, the Renaissance Era when Frigates become available. With a tiny navy, even the absence of Mine Fields won't see a city fall. Have you seen how much health those cities have late game? By the time a small navy gets a city to yellow, WW is going to start kicking in and it's harder to maintain the siege. If they don't change this fact except making games longer, why are we keeping them?
1) Naval Invasions are some of the trickiest military strategies to do historically, and that carries into the game as well. This is especially true for the AI, while the AI is murder on land, it still really struggles to beat its army on your shores, in a way that doesn't just get them murdered left and right.
When people talk about runaways, its often the "runaway on the other continent". The reason for that is because....its simply harder to project force to another landmass in the game right now.
So the key question....is it too hard right now? Removing Mine Fields would make it a little easier to do invasions. I don't think its completely changes the game, but it is a step in that direction.
I personally think Mine Fields make it slower, not harder, to do invasions. I can still use my same strategy of patiently picking off enemy land units with my Battleships until I'm bold enough to proceed. Instead of 10 turns, I might require 20 turns. The outcome doesn't change because the AI will still struggle to defend without its navy which I'd likely crushed before I enter water controlled by Mine Fields.
2) Should late game reward offense or defense? Historically there are periods of warfare that favored defense (medieval times, WWI to a certain extent with "trench" warfare), and then periods where offense is king. In modern warfare, offense is the name of the game.
Is naval offense rewarded enough in the late game? Should a player with a strong navy be able to take the cities of a turtle player without a navy or air force? (we need to keep in mind an air force is a strong counter to navy at this point in the game...and interception is not what it used to be)?
If the answer is yes, it may make sense to remove mine fields.
We generally start on the defensive and then shift to offensive. Why? Defending is usually easier and, if I can crush the enemy navy in my territory, I can repair all the damage and proceed to attack afterwards. In essence, this isn't merely my naval offense. This is a combination of both to shift the balance in my favor as I eliminated some of the most dangerous enemy units (like Destroyers) and let my Battleships have free reign over the enemy.
Personally, I do feel that, after all the time we invested to reach the Atomic Era, we should avoid any mechanics that slow it without any good reasoning. If your navy is strong enough, a Mine Field isn't going to change the outcome. It will increase the turns you need to achieve the same thing and that's, in my opinion, very anti-fun.
3) If mine fields had never been a thing, and someone proposed them today....would we want them?
This is a question I think is very important to continuously ask as the mod develops. Its easy to use inertia and say "well its been there so keep it in". But that's a way for glut to clog up your game. So if mine fields didn't exist, would we have a late game "problem" that we would be calling for something to solve? Would people say "late game navies are so OP, they are unstoppable!" ?
If the answer is no, than should the building exist?
I think the question "late game navies are so OP, they are unstoppable!" can be used to addressed even the Renaissance Era. Corvettes and Frigates are in a really good spot as most of us agree. With one extra movements, a Frigate can attack a city without taking any damage since it can retreat three tiles away from the city. However, nobody is complaining about this. Then, we have the Ironclad and Cruisers. While Cruisers have to work much harder to take no damage from cities, it's certainly doable and, without an enemy navy in the way, they also dominate in their time. Why then does Atomic Era make a difference with Destroyers and Battleships?