If you DO find that one city state, then you are guaranteed to enjoy your Swordsman rush, as the AI will not have 5 envoys to counter your Mysticism-Amani-first levy combo. Sure, they might be some incidents in the later game, but you have basically won the game after your first rush, so you should be able to cope with the issue. As for Amani, the opportunity cost of choosing her is actually not that heavy - some quite good players even prefer to start with Amani in non-Hungary games as she is able to score 6-9 era points easily for a classical golden age.
This was my experience with Hungary as well. It is possible sometimes to start without any city states on your starting continent (with a land connection to walk along [pre-embarking] to aid your troops). However, that seemed to be quite rare and less situational than 'other situational' things that affect many other civs in the game.
The AI does often go after the CSs early. However, even in one game when my AI neighbours were gobbling up the CS around me (Scythia and Ottomans), all I needed was just
one to get the ball rolling. That first levy... I even freed one CS 'normally' before getting my first levy and got two free envoys in the process but it didn't have an army to help until much later. Taking Amani as your first governor, so long as there is one CS left that you can use, made this quite consistently achievable. But yes, you need that first levy to happen... Even getting it before Iron Working is still nice, but of course the notable example now is beelining that tech and upgrading a bunch of levied warriors into a ridiculously-early Swordsman army with movement AND combat bonuses that will steamroll anything in its path.
I really like Hungary but now I wish they were designed differently. My wife and I have started playing hotseat games sometimes when the kids are asleep (we're not into online multiplayer yet and we like to play on Marathon speed on continents maps), and Hungary is now '
banned' as a choice. We both like having a chance to get at least some CS suz. in our games.
I wish they just had a different bonus to getting envoys or maybe even a free diplomacy card slot for having an edge towards getting suz. instead, if they wanted to use it that way. 'Buying CS envoys' is too easy and you can just keep doing it as soon as a CS is available for its troops to be levied again. I had lots of unused envoy points earned the normal way 'just in case' someone managed to take a CS away from me. I didn't need to use them normally. I controlled every CS in the game and any that had been conquered I liberated. Controlling every CS in the game is a very strong position even if you never go to war again.
I like that Hungary made me take a closer look at levying troops. I now sometimes use it with other civs as opposed to never touching it.
As for levying troops, I think that's great as a part of their flavour. I just wish it wasn't so strong. I felt like I was cheating at one point playing Hungary and it started to feel like I won as soon as I got the ball rolling.
I also would prefer my 'elite troops' to actually be 'my troops' and not the mercenary cannon fodder I levied. In reality it became the other way around. I think Hungary could retain their flavour if levying was instead easier for them
but not pumped up on steroids so much.
• Maybe give them a discount towards levying troops that can stack with the government plaza building that also offers a discount. Maybe make it so that it can be easier to do but without the same "Surprise Moth%$&#^*!" pumped-up troops with free upgrades thing in play...
• Let the levied troops be normal levied troops. Take away the free upgrades and the bonuses they get. This more than anything is what is so strong (later it's also that it's effortless to liberate and control every CS in the game). Let them be useful cannon fodder and let the Black Army UU shine more.
• Take away 'buying envoys' and replace it with some other kind of edge towards getting envoys. Something good, but an edge where's there's a chance for other civs to compete for CS beyond just wiping Hungary off the map.
They are not the only cool and fun civ that seems out of whack in terms of balance but they are one I've played recently.