Don't worry, plenty of us here played it for years, if not decades. 5e is a lot like 2e AD&D (and to a slightly lesser extent, since they weren't that different, 1e and the early game) in a lot of ways. Combat, oddly, not so much one of them, in significant part because of bounded accuracy. No wonder you weren't making much sense, if you thought BA was a feature of the classic game. BA is actually closer to the 4e 'treadmill' than to any prior editions' scaling mechanisms.
The similarity is more in the classes, and the swinginess at low level, and - most importantly - the degrees of freedom left to the DM.
Yeah, 5e combat is kind of its own thing. Its swingier and less tactically elaborated than 4e, but far less so than AD&D. It certainly isn't as much rocket tag as 3.x but it can get pretty dicey at higher levels.
As for the bonus progressions of different games...
In AD&D a fighter gets +1/level (or +2/2 levels, whatever) up to needing a 20, which then repeats 6 times in 1e (but not in 2e). Add in expected increase of +3 from magic weapon and +3 from other magic (gauntlets, etc) and you get something like maybe roughly +18 over 12 levels. 4e is built on a +1/level math engine, but level 20 is more like AD&D level 12, so you might consider it to have basically the same progression, or less if you just compare levels 1-to-1.
5e obviously has a much slower progression, I think it was roughly about +1/2 levels, but I haven't played a fighter, so...
Of course HP and AC progression are also important. AD&D has little progression of AC, relatively. A level 1 fighter might be AC5, and he might be AC -5 at level 12 (but probably more like -2, depends on the game as it 100% depends on magic). 4e obviously progresses AC the same as to-hit, and 5e is closer to AD&D. So generally in 1e and 5e you'd expect higher level PCs to hit more often. In both of these games though HP progress at a steeper rate than in 4e, so 4e characters take less hits at higher level but have relatively less hit points (without getting into healing, which changes things a lot). 3.x here is just wonky, with damage mushrooming much faster at high levels, but SODs become the most important thing anyway, so the progression doesn't really matter.