Sure..
You would reduce the numbers of responsibly bred dogs..
Reduce the gene pool of certain pedigree dogs to a dangerous level..
To work you would need serious punishment for those breaking the law and breeding during a ban or breeding without the required permissions... only really heavy fines or confiscation of the dogs would actually be a deterrant.
We do not have the infrastructure required for licencing breeding in this way, nor for policing it - the cost would be enormous adn it would only work if it was run by central government. Local government already have responsibility for licencing commercial breeders ... they do a DIRE job of this, hence the number of puppy farms.
The result would be that within a very short time we would..
Lose many pedgree breeds entirely.
Seriously damage the ones we don't lose.
More of the available dogs would be produced by illegal or disreputable breeders than they are now (and as it stands, they are the majority right now).
Temperament and health would be reduced even further than it is now, we would see a rise in dog attacks, heavy legislation on what dogs can do, a further knock on effect on behaviour as dogs are not permitted to be off lead, socialise, etc...
It is a horribly bad idea, all round.
Reputable, responsible breeders are NOT adding to the problem, the dogs they produce rarely end up in general rescue and when they do, many breeders try very hard to get them out (and many are thwarted by general rescues who refuse to interact with breed rescues!).
Good breeders ask that owners relinquish dogs back to them at any point if the new owner cannot keep them - for this reason you will rarely see breeds like deerhounds, GBGV, PBGV, Afghans, Borzois etc, in general rescue... yes those are rare breeds but they have healthy numbers... the breeders and breed clubs are like the fecking MAFIA... and they deal with their own 'mess' as it were.
Two things need to happen:
People need to stop buying on a whim - it is that demand that drives supply.
People need to see the value in a well bred, well raised puppy produced by a dedicated and ethical breeder.