Still haven't finished modern age, but played through antiquity and exploration a couple of times (on Sovereign).
Antiquity feels pretty much natural. You are getting progress for the things you would naturally do anyway, only if you want to reach the last milestone, you need to focus on the specific things, that may contradict something else.
Economy: you need to focus on trading and creating cities + building markets in order to have all the resources and a place for them. Otherwise, you will probably safely secure two first milestones.
Culture: Wonders. Again, you need a crazy amount of wonders to pick the last milestone, so you need to focus on going "deep" on both culture and science while playing tall and using excess gold to buy the production buildings. Otherwise, you'll probably get a first milestone from a couple of wonders you naturally get from beelining important things in the tree. It must be noted, that AI loves to build wonders and they are pretty difficult to get.
Science: You will naturally get some codices. You will also get some from mastery techs you need, but if you want to hit the last milestone, you will need to go "wide" with research and get as much masteries you can.
Military: Just expanding up to your limit will give you a couple of milestones, but to get to the last one, you will probably need to capture at least 3 cities.
Exploration, on the contrary, fails to be like that and feels like a separate game. You need to pursue the very specific goals and in a specific way ignoring the natural way to play (and also very eurocentric, but this seemed already inevitable)
Economy: you have to settle on another continent near certain luxuries. This is it. Your valuable settlement limit should go there instead of a plenty of territory that may be available nearby.
Culture: you need to build missionaries and spread religion in order to get relics. The ones you will get from the research trees are probably not enough even for the first milestone, but I haven't counted. Religious gameplay also far from ideal and your founder belief determines how much will you suffer while spreading religion. In addition, some of the best civics in this era require you to maintain your own religion in your cities. So you are essentially being pushed into spending production on missionaries.
Science: I think that the developers just ran out of ideas. There was nothing about middle ages that screamed "science" to them, so they went ahead with... tile yields? Anyway, even the given solution is weird and requires certain counter-intuitive planning. Also the best city buildings to achieve this are probably the coastal ones that give you bonuses to food/gold as they have the better initial yield and better adjacency (if someone managed to get 40+income in the middle of the continent with, for example, mountains and wonders, please let me know. I think this should be possible, but you really need to work on surrounding a future city tile with wonders.
Military: requires you to colonize the distant lands and bring your religion here. So, basically, all the disadvantages of economy and culture together, yay!
So, yes, exploration is really not about "playing in your own way"
How would I suggest to fix it (please stop reading here if you are not interested in some amateur opinions)
Economy: I would probably add a concept of "far frontiers" starting with the antiquity. Make tundra and equatorial territories (jungles, deep desert) impassable in the ancient era. This will make the continent more cramped (it definitely has too much free space right now) and will give a land "exploration" option. Unique resources found in the far frontiers would generate "caravans" the same way as trade fleets. Maybe slower to outbalance the distance.
Culture: First of all, there should be a "religious isolationism" social policy which should give make it impossible to send your missionaries to other civs, while also closing your borders to outside missionaries, so you could eliminate yourself from this completely. On the other hand, there should be "Great works of art" that can be generated in some other way. Maybe the number of specialists should contribute to "Great works of art" points, modified by natural and world wonders in the city. To balance this out, maybe the number of relics from the civics should be toned down.
Science: The existing solution should be completely thrown in the window as it doesn't make any sense. What should go well with the "exploration" topic is making "discoveries and breakthroughs". Each discovery should give some sort of small benefit with "easier" discoveries belonging to the distant lands, while others will be focused on domestic development. For example: find a new natural wonder in distant lands, work on a new treasure resource, be the first to build a certain number of hospitals (medical breakthrough), be the first to have a certain number of arquebusiers (chemical breakthrough) and so on.
Military: I'd say military should stop putting all points into distant lands. I wold suggest giving a point for a city in a distant lands (double if captured). One point for capturing a city in a homelands (double if converted). Four points for capturing an opponent's holy city.
Thank you for reading if you got this far.