Yup. Building wealth through home ownership directly implies homes becoming unaffordable for new buyers.
The point is that there isn't a distortionary corporate investment force preventing young people from doing that; what there are is zoning and permitting rules locking every desirable location into stasis.
* Subsidizing first-time buyers will increase home prices. That's the opposite of the point.
Subsidizing first time buyers does increase prices overall but you have to consider the effect of such a policy relative to its cost. It has potentially positive externalities so that’s worth considering while balancing it out eg with the previous policy.