Is Three No Trump ever better than a four-four major fit? Yes, but it is not good as often as players try Three No Trump.
People with a flat hand, a hand that is four-triple-three tend to bypass Stayman because a hand with no ruffing potential is likely better in No Trump. That is partially true. If both hands are four-triple-three and all suits are sufficiently stopped, then Three No Trump will score better than a four-four major fit. However, a short hand ruff will make a major contract better than No Trump and that short hand ruff can be taken in either hand.
However, one does not know if partner has a short hand ruff or has a suit insufficiently stopped. Therefore, with a four-card major and eight or more HCP’s, one should always use Stayman.
The Bidding: South, with a balanced 15 to 17, opens One No Trump. West has a decision to make. Should he be quiet and lead the Club Jack against a hoped-for Three No Trump in to which could be declarer’s waiting Ace and Queen or should he interfere? If he interferes, partner may raise the suit in a part-score battle or a sacrifice. If the opponents win the contract, leading the suit can be left to partner.
West, using DONT, doubles showing a single-suited hand. North redoubles showing a good hand and guaranteeing values but wants to wait to hear what the suit is. East does not need to bid the relay Two Clubs because North redoubled. West bids Two Clubs and North (playing Lebensohl) bids Three Clubs which is Stayman without a stopper. East doubles for a Club lead and North and South find their Heart fit.
The play: West leads the Club Jack and East wins the Ace and returns a Club. Declarer ruffs the third Club and draws trump. South loses two Clubs and a Diamond making Four Hearts for +620.