The problem with two weapon fighting is you need to use your bonus action all the time to get the most out of it, which makes it a problem for Barbarians, Bonus-action intensive Fighters (Rune Knight) or Rangers who have a lot of bonus action spells or abilities.
Yes... that's why I said you should make it cost the user's Bonus Action. Several classes and subclasses don't want to give that up. As a result, this "Extra Attack +1" feat would be directly comparable to Dual Wielder. You would be trading 1 point of AC and the other minor benefits of dual-wielding for getting 1 additional regular attack, presumably using a heavy weapon. That means, at absolute most, you'd be looking at (hit rate)×(1d8) extra damage per round vs (HR)×(2d6 w/GWF) extra damage per round, factoring out ability modifier since those should be the same (abbreviating hit rate as HR.) Roughly (HR)×(3.83) extra damage. Champions would get (hit rate+0.05n)×(3.83), where n is either 1 or 2 depending on level. So, for a typical attack that has around a 65% chance to hit, you'd be looking at almost exactly 2.5 extra damage per round, or 2.87 per round for Champions.
By comparison, getting GWM or SS would give you +10 damage per successful attack, but -20% hit rate. Having crunched the numbers, when you can only make 1 attack, "Extra Attack+1" is worth 6 extra damage per round. As N goes up, the benefit of "EA+1" goes down, to a minimum of 1 damage per round at 4 attacks.
So, yes, this would start out contextually better than GWM/SS at very early levels, but quickly becomes essentially identical: doing ~3 extra damage (on average) for an
entire round is not a massive difference at level 11. Ironically, having low chance to hit is somewhat beneficial to GWM: the closer to maximum hit rate, the more damage "EA+1" gets over GWM, but this also causes the two to converge more slowly as N increases. High hit rate puts "EA+1" ahead early on, but quickly erases or even eliminates all benefit once you get to N=3 or 4.
Thus, my proposed restriction of requiring a moderately high level (my original proposal was 11, which is when the gap between "EA+1" and GWM essentially vanishes),and costing the user's Bonus Action to activate, seems to be perfectly in keeping with other powerful feats like GWM.
Note that I technically left out crit damage for both things, so the numbers above are very,
very slightly off. But not by much. Edit: At 1 baseline attack, 65% hit rate, "EA+1" is worth +7.25 average damage per round, accounting for crits, which is pretty high at that point. This falls to ~5.42 average damage gained per round (not per attack, per
round) at 2 baseline attacks, ~3.58 bonus DPR at 3 baseline attacks, and 1.75 bonus DPR at 4 baseline attacks.
Furthermore, note that GWM itself
also grants a bonus action extra attack, albeit under conditions (critting or reducing an enemy to 0 HP.) So there's at least a portion of the time where GWM gives you everything "EA+1" does
and also the optional extra damage.
So...yeah. This is not a crazy strong feat (after level 5-6, once martial characters all get at least 1 EA.) It's just not. If this paltry amount of bonus damage per round is horrendous, then numerous other parts of RAW 5e are brokenly powerful already. When enemies may have hundreds of HP, which isn't at all unlikely once you hit level 11, doing less than 3 extra damage per round more than GWM is laughable.