Are you certain? Congress has recently appropriated a 10% increase for two border security agencies, so are you sure that none of the funding was earmarked for more detention capability?
Once you start down this path of the locals taking care of it, your right back to the problem of the feds re-imbursing the state/local for these costs. The record on this is not too good.
Why is the record not too good? Is it the fault of Congress not appropriating it, or the executive agencies not being forthcoming with it for some reason?
As for why Congress has always been dragging feet in re-imbursing the border states for the cost that they incur, this mainly boils down to congressmen from other states not wanting to spend the money. So as the illegals show up around the country, they will want their share also. If Texas gets re-imbursed, GA and NJ will want to be re-imbursed also.