Can you choose your carrier when a carrier owns the data centre?

I've chosen such datacenters because of the carrier

Usually because we have customers of that carrier complaining about speed so we stick some servers for a CDN / Reverse proxy farm.

Yes but

Having worked for a number of carriers (ISPs), DC owners often contacted us asking us if we would like to install fibre and a POP into their DC. Of course they don't want to pay for this, its "an opportunity" for us to compete for their clients business, to recoup our investment.

Lots of this happened in the 90s and early 00s but not any more because the returns are either terrible or you make a loss. DC owners generally need 2-3 diverse fibre carriers minimum for credibility and they usually have to build this into the budget during construction.