Balanced over the Unbalanced

K 10 5
9
A Q 10 8 7
9 8 7 5
A 7 6 3
A 4
K 6 3
Q 10 6 4

Playing in three clubs, South took the opening lead of a heart, ruffed a heart in dummy, then pushed the 9 of clubs through, losing to the jack. A diamond was led, captured in the closed hand, and declarer led a second round of trump, both opponents following, East winning with the A. Declarer took a second diamond return in dummy, and now with only the high trump out, simply ran diamonds, sluffing two spades. It didn't matter when the holder of that last trump ruffed in. South had both the controls and entries to sluff two spades on the long diamonds. I.e., he did with that defense. Had the defense gone after spades, it would've been a different story.
So declarer lost only those three trump tricks, making 4. East remarked that diamonds would have served as well, but South pointed out that no, in diamonds he would have no sluffs and hence would make only 3. (Diamonds would actually have been the third best denomination, since declarer could run 8 tricks in no trump, ironically completely bypassing the suit that works best as trump!)
Now this was a partial and the 20-point advantage might seem trivial, but there are several counter arguments to that. For one thing, the next hand offering such a choice at that level might be a matter of hearts and spades, and then the difference between 9 and 10 tricks is not so trivial. Or if you add in the jack of clubs, now a minor-suit game is a reasonable risk in the balanced suit, but has no chance in the other. Nor can that difference be considered trivial in a matchpoint game. So in general, it behooves you to name the balanced suit trump for the profits that may be there.
The hand also illustrates my contention that "game" is the touch-and-go level where advantage may or may not reside in the balanced suit, but IF there is an advantage, it will almost surely lie with the balanced suit as trump. Four clubs isn't game, but that's academic since that level would be game in a major. So call it the 9- or 10-trick level if you prefer. Had the defense knocked out a spade stopper on the opening lead and then again on the first regaining of the lead, they could cash a spade when they next regain the lead and there would prove to have been no advantage to the clubs as trump. But they didn't find a spade attack and declarer was able to take advantage of it. In diamonds, it wouldn't matter whether the defenders found the spades early or not. They're always getting a spade trick.

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Q 10 8 7
A Q J 8 6
K 10 7 5
A J 5
A 9 6
K 9 5
A Q 8 4

This hand is from a nationwide tournament. It was played variously in 6 no trump (going down except for one time, honest!), 6 diamonds, making because the king of hearts was favorably placed, and 6 clubs, making with an overtrick without any concern for the king of hearts! Which is to say that with an offsides king, you can't make little slam in diamonds, while clubs offer grand slam!
How is it that we not only get an extra trick in clubs, but don't have to concern ourselves with the king of hearts? The reason is that you can ruff two spades in dummy for a total of six club winners, to go with 5 diamonds and two aces. Well, can't we ruff spades in a diamond contract? Yes, certainly. But those ruffs are part of the 5-card length of the diamond suit and don't add to the number of diamond winners. They're not substantially different from running five diamonds sluffing two spades. So you have five diamonds, 4 clubs, two aces for eleven! And so you need to find that king of hearts onsides.

K J 8 7 5
K 10 9 7 6 5 4
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A
A 9 6
A
Q 7 5 3
9 8 6 5 4

WestNorthEastSouth
1 2 * Pass 3
Pass 6 AllPass
* Michaels
The vagaries of a Michaels cue bid (showing both majors) made South declarer in this six spade contract. Declarer ruffed the opening diamond lead, came to his ace of hearts, went to the ace of clubs, ruffed a heart, everyone following. When he now laid down the ace of spades and continued the suit, the queen appeared second hand, so he wound up making all 13 tricks. He didn't absolutely need that 13th trick, but in a heart contract, he makes one trick less, since he can't ruff out the second round, and picking up the queen of spades then becomes the all-important matter of making the contract or not.
Now this is not a certain advantage awaiting you with balanced suits nor even an altogether probable one. It will most likely work at the slam or near slam level, where you are likeliest to have the controls and the ability both to run trump and to set up an unbalanced side suit to get sluffs. Thus it was no accident that the most conclusive hands above were all at this level (on the three club hand, the defense could have knocked out the spade stoppers and won a spade trick playing double dummy.)
At the game or near game level, the advantages are much more tentative. Still, they are occasionally there, and you will want to prefer the balanced suit so as to be ready for those potential advantages when they are there to be had (not to mention developing the techniques for finding your balanced fits on stronger hands).
I take this hand to be fairly representative of the possible but by no means certain advantage of the balanced suit as trump at game level.

9 8 4
K Q 8 6
A 9 8
K 4 3
A K 7 6 5
A J 9 5
10 6 2
6

Now if both major suits break 3-2 and you get a diamond opening lead, it's not going to matter which suit you're in, barring some fortuitous blocking. You're going to have to lose a spade, two diamonds and a club. Yet hearts are definitely to be preferred as trump. Why?
One reason has already been suggested: you may not get a diamond opening lead. Now in spades, you're going to need an on-sides ace of clubs for this to be an advantage, which may or may not be there. But in hearts, after three rounds (we are talking 3-2 splits here), you'd lose a spade and now it's too late for the defense to hit diamonds. Upon regaining the lead, you'd run spades, sluffing two diamonds, which doesn't mean you can ruff two diamonds, since there's only one trump left in dummy, but it does mean the defense can take only one diamond, along with a club and spade. I might refer to the possibility of blocking in diamonds. Even the K J would constitute a partial blocking. Though the hand with the Q could overtake the J, declarer's twice-guarded 10 would mean that that doesn't help the defense. So even a diamond lead (if declarer takes the first round) wouldn't mean the loss of two diamonds in a heart contract.
Also, if we translate the king of clubs into either the queen of spades or king of diamonds, then it would make a difference whether we chose the balanced suit or not, whatever the opponents did. In those circumstances, game would be almost routine in either major suit (because the "king of clubs" has been made a solid winner), but now you would be going for an overtrick with the balanced suit as trump, for the reasons given.
Of course we can't literally translate the king of clubs into another card. But as you bid, you can't know where your partner's points are going to lie, and IF there is an advantage, it's going to lie with the (presumably adequate) balanced suit as trump, the unbalanced suit on the side.
Here's another game-level hand I chanced on just yesterday on OKbridge:

9 8 7
A J 7 5
A 10 7 3
A
10 6 5 4 3
K Q 8 4
K 5
K 3

I didn't memorize all the spots, and I'm not a 100% sure of the bidding, but I am certain about the distribution of the major suits and the clubs, so the principle is manifested whatever my uncertainties. North opened with a diamond and East pre-empted with 3 clubs. A negative double would doubtless have steered N-S in the right direction. But South chose to bid 3 spades, which was raised to four by his partner, making easily on the 3-2 split.
But look at the advantages of a heart contract. First, a 4-1 spade break doesn't automatically doom you in hearts. Also, note the clubs and the potential for sluffing a spade from the North hand, perhaps holding your spade losers to two in a heart contract. You're going to need a little help for this to happen. Obviously if the defense starts off with three spade winners, you're not going to sluff a loser. And if you get a 4-1 heart break, it's probably going to give you as much trouble as a 4-1 spade break would have. Further, you need spade winners, not an absence of spade losers, and you might find it difficult to get 10 winners without developing spades. Still, it's not impossible that you'll be able to finagle an extra trick in hearts as trump that you can't pick up in spades. This again demonstrates how at the game level, the advantage of the balanced suit is not certain or even probable, but likeliest to lie with the balanced suit as trump if there is a difference.