Here's a simple fact. Densification never leads to lower prices. As you build more housing, it _always_ becomes more expensive faster than you can build it.
I analyzed the database of all real estate sales in the US for the last 25 years, and I have not found a single example where densification led to price drops. Other scholarly literature found similar results, at most you can expect single-digit percentage drops on rents from new construction.
> There's nothing wrong with letting urban centers densify
Yes, there is. Densification forces jobs to migrate to urban centers, because companies in dense cores have a competitive advantage in the access to a larger pool of talent.
So your choices are:
1. Densify to hell and then in a couple of generations your children will live in apartments where you can shit into your toilet, while cooking food at the same time. See: microapartments in Tokyo.
2. Limit density and promote suburban lifestyle.
That's it. It's not a false dilemma, there are really no in-between choices that are stable long-term.