Why would adding a new supplier to the market cause the price of power to go up?

Because on-site powerplants owned by datacenter operators are not "just another supplier".

The threat is: This "datacenter power" disincentives buildout of "free" powerplants (by eating up significant demand at very low margins thanks to basically vertical integration); this slows down buildout of "normal" infrastructure (possibly both grid connectivity and power), and the electrical energy market becomes worse for consumers than it is now.

I personally think all of this is very speculative for now, but allowing industry to rely on the grid (which they still would!) while almost exclusively "buying" their own power is a risky proposition from a consumer perspective.