Your particular suggestions need work.
You also need to watch out for other forms of multi-attack, like TWF, or crossbow expertise.
Last thing that 5e needs is less attacks.
I would put for every martial class extra attacks at lvl5,11,17, same as cantrip damage boosts.
Then give fighters one extra at 20 and some other ability at lvl11.
And martial classes levels would stack for those extra attack.
One attack per round is god damn boring.
I disagree with the premise of this post.
I think extra attacks are so integral to the nature of the martial character that I wish they would bring back the "attack for every level against less than 1 HD creatures and non-exceptional scrubs" rule from AD&D.
20th level fighter? 20 ATTACKS! Muahahahahahahaha! Now you're cooking with gas.
I do have "split attack" for attacking multiple targets for that reason. But, a big part of me lives Book of Nine Swords and wants "martial fireballs" for chewing through hordes of mooks too.
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Would you take away things and leave "dead levels" at those levels, or would you give lesser abilities so there are no dead points and accept that weapon damage is going to get a buff across the board (not just for non-fighter and non-rogue)?
I could get behind universal Increased number of attacks coming back in some fashion. I want universality in one way or another, both to smooth out multiclassing and to unify game elements.
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If you have the extra attack class feature you can forgo using that feature and instead make a single attack when you use the Attack action.
If that attack hits, it deals an extra +3d6 damage (of the same type as your weapon) for each extra attack you could normally make with the extra attack class feature.
You must choose to use this feature before you make your attack roll.
You may also use this option when you ready a single attack using the ready action.
See that is a lot better option than the OP idea and plays nice with things like Champion and classes who get a 2nd attack but with no weapon style.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.