I did a lot of thinking and simulations about bidding after 2NT and I am about sure that:
a)most systems people come up with are much worse than the simplest possible
b)searching for 5M in opener hand is not important, especially at imps (you may be surprised about this but there are a lot of 10-9 deals (10 tricks for 4M, 9 for 3NT) and getting fast to 3NT makes them to misdefend often, especially on first lead on 8-9 or 8-10 hands hence it's more important at imps to just bash 3NT than it is at MP's).
c)it's important to be able to invite slam holding 6M
d)it's important to have a slam try in minor available
e)it's important to be able to show any 5-4 combination (which is general principle of bidding opposite balanced hands)
This leads us to following system after 2NT:
3
♣ - normal stayman promising at least one 4M, then smolen or w/e for 5-4 majors; if we fancy we can add 3NT/4C responses for 4-4 in majors and min/max with retransfers to follow.
3
♦ - transfer with superaccepts, after normal accept other major is slam try in M; tranfer and 4m is natural slam try 5M-4+m (and then 4NT is to play, and one other bid should be 6 card RKCB); if one partner shows 5-4+ opposite balanced hand it's good to have 6 card keycard in your system.
3
♥ - as above
3
♠ - minors, at least mild slam try
3NT = to play; this is very important, any system which has this as something else is bad. No engineering could overcome loses from not being able to bash 3nt and having them guess first lead
4
♣/4
♦ = natural slam try (then 4NT to play, others are cuebids or kickbacks depending on what you like)
4
♥/4
♠ = w/e you fancy
With 5-5 majors you bid stayman, then you have tons of options: smolen and 4H, smolen and 4S, smolen and 4m, instant 4H, instant 4S. You can accomodate every range/splinters/invites you fancy. All of those:
2NT - 3
♣
3
♦ - ????
1)3
♥ and after 3NT: 4c/4d/4h/4s
2)3
♠ and after 3NT: 4c/4d/4h/4s
3)4
♥
4)4
♠
Might be used for various hands with majors. It's enough space for most scientists. Maybe even some of them could be used for 5-4-3-1 shapes and slam invites. Or maybe 4H/4S could be used for 4-1-4-4 hands or w/e.
That could be improved by various things like:
2NT - 3C
3D - 5D = 4-4-0-5 which never comes up but make all shapes biddable
This doesn't require memory work and I think it's better than anything else available as of now. I went trough tons real and simulated hands to prove it to myself.
About the only leak of this system are (3-1)-5-4 hands good enough for game or stronger. This is small leak because:
a)those hands are rare
b)it's even more rare to find 5M in partner's hand fitting our 3 card fragment
c)if we don't have 8card M fit then we are much better of than puppeters. After puppet they know something about clubs (dbl or not) and something about opener majors; they will make much better 1st lead decisions with that information.
Gains from c) are significant and frequent. They are more valuable than rare loses from missed 4M contracts.
In fact, 5M-3-3-2 in 20-21 range opposing 3-1-5-4/3-0-5-5 resulting in 8 card M fit is so rare occurrence most bridge player will never see it in actual play during their entire bridge life.