More housing, regardless of type, is directly correlated to cheaper average rents [0]. So Thiel is right in that regard, the red tape for construction and opposition from NIMBYs must be curtailed.
NIMBYs typically oppose especially lower cost housing, so it's not surprising that cutting red tape does help reduce housing prices, or at least slows down the rise.
The question is if it is enough - and I would be quite certain it's just a band aid. Unless you impose the construction of cheap housing, cheap housing will tend to not get built - almost every interest is opposed to it.
Good thing Mamdani is doing both, cutting red tape and constructing new housing, at least that's his plan, we shall see what others in his governance think about it.
[0] https://www.journals.uchicago.edu/doi/full/10.1086/733977