The initial version of the question gives 14 cards to South. Assuming the hand is actually AKx JTxx Kxxx Ax.

First, let's explore whether the hand you have given plays better in 3NT or 4H:

You have 7 top tricks (3 spades, 1 heart, 2 diamonds, 1 club). At 3NT, you must develop two more tricks, and hearts seems like your best shot at that. You must hope for honor-doubleton on your right (run the jack; if RHO covers, win the Ace and play up to the ten. You will lose to the honor on your left and then your ten and long heart are winners). If the honor holding is unfavorable but the split is 3-2, you still have hope of picking up the fourth round of diamonds (duck a diamond, then play the A and K; hopefully they have broken 3-3). You will have held up the CA on the opening lead. If clubs broke 4-4 then you'll make if you were able to take 3 hearts but go down if you lost two hearts as you still must lose a diamond. If they were 5-3 you're always going down on a club lead (and if hearts didn't behave, the opponents can still defeat you with a club switch). If they're 6-2, you can make in the unlikely case that the 1-3 tricks you must lose before taking 9 tricks are all lost to the doubleton club holder.

In 4H, you have a sure club and heart loser. In order to develop ten tricks, hearts must break 3-2 and you must ruff a club and a diamond, which requires losing a diamond, so you must be able to play hearts for one loser (which means Qx or Kx on your right). However, you make even if clubs don't break 4-4.

So this hand isn't a great example of a hand that would want to avoid Stayman. That being said, there are times when jumping to 3NT with a 4-card major is indicated. Here's such a hand:

AKx KJxx Q9x JTx

This is a flat, meaty hand. You don't have ruffing values, but your side has a combined 29-31 HCP. Not enough for a slam, but very comfortable for game -- unless opponents can establish a ruff or two before you gain control. Let's give partner (South)

Qx AQ9x KJTx KQx

You'll almost certainly take 11 tricks in NT. On the other hand, in 4H, it may easily go diamond to the A, diamond ruff, club to the A, diamond ruff.

In the hand given, you can sometimes make 3NT and never make 4H without misdefense. But change South's hand slightly to AKx JTxx Kxxx Ax and 4H has more chances than 3NT:

You have 7 top tricks (3 spades, 1 heart, 2 diamonds, 1 club). At 3NT, you must develop two more tricks, and hearts seems like your best shot at that. You must hope for honor-doubleton on your right (run the jack; if RHO covers, win the Ace and play up to the ten. You will lose to the honor on your left and then your ten and long heart are winners). If the honor holding is unfavorable but the split is 3-2, you still have hope of picking up the fourth round of diamonds (duck a diamond, then play the A and K; hopefully they have broken 3-3). You will have held up the CA on the opening lead. If clubs broke 4-4 then you'll make if you were able to take 3 hearts but go down if you lost two hearts as you still must lose a diamond. If they were 5-3 you're always going down on a club lead (and if hearts didn't behave, the opponents can still defeat you with a club switch). If they're 6-2, you can make in the unlikely case that the 1-3 tricks you must lose before taking 9 tricks are all lost to the doubleton club holder.

In 4H, you have a sure club and heart loser. In order to develop ten tricks, hearts must break 3-2 and you must ruff a club and a diamond, which requires losing a diamond, so you must be able to play hearts for one loser (which means Qx or Kx on your right). However, you make even if clubs don't break 4-4.

Consider some other hands declarer may have:

Ax KJTx KQxx Axx

KJxx KQxx KQx Jx

xxx QJTx KQx AK

All of these have more play in 4H than 3N.

So this hand isn't a great example of a hand that would want to avoid Stayman. That being said, there are times when jumping to 3NT with a 4-card major is indicated. Here's such a hand:

AKx KJxx Q9x JTx

This is a flat, meaty hand. You don't have ruffing values, but your side has a combined 29-31 HCP. Not enough for a slam, but very comfortable for game -- unless opponents can establish a ruff or two before you gain control. Let's give partner (South)

Qx AQ9x KJTx KQx

You'll almost certainly take 11 tricks in NT. On the other hand, in 4H, it may easily go diamond to the A, diamond ruff, club to the A, diamond ruff.