It's a good start. Would be nice to add a gradually increasing tax on multiple home purchases, e.g. buying a third home has a 5% tax, 4th 15% tax, fifth 25%, and if you want a sixth home you have to build it yourself. Would prevent regular people from buying up dozens of homes to rent.
It should be much more punishing from the 4th home like 45%. One can understand 2/3 homes, e.g., one to live in and the rest as investment for retirement income. More than that is just greed. Also, if you don't live in one of your properties, either sell one to keep the total number within 3 or pay additional tax.
Changing the excessively huge (and probably financially needed) capital gains deduction for selling a home and buying a new one to only work if you literally buy new construction - would probably spur something.
Existing stock would be depressed in value as the pool of buyers would be smaller, but they'd be more likely to be first time buyers anyway (no capital gains to deal with) and demand for new would increase.
But you could end up with insane inversions where existing stock was pulled down to build new to take advantage of the gains exclusion.