I guess this would be similar to the way a lot of sports arenas are built. The public foots the bill to build it, and the team gets a cheap 20-year lease to use it. And the public the gets a sports team.
The public foots the bill, and the owners of the team profit handsomely from selling the public the chance to watch the team play in the stadium they already paid for.
Sports arenas built with public money are a scam, and so is the idea of having taxpayers subsidize the construction of for-profit privately-owned chip fabs.
It's a voluntary exchange. If Intel's requirements for participating don't work for the locality, nobody is required to participate.
If the locality chooses to pay with public funds, that's their decision to make. But it doesn't immediately erase the history and override the terms of the exchange that was made.
The parent was asking why public funding implied public ownership. That is the default. Multibillion dollar negotiations are a series of complicated compromises, so that the public would be willing to finance a fab they will neither own nor hold power over to satisfy global demand is well beyond the 'burden of proof' line.