While that's true, it's ineffective as a defense.

First, the fact other shelters are willing to overpop to take animals in need and are generally not overpopulated right now. Intake counts have been stagnant and predictable year over year. Shelter and adopter supply exists to match the demand and PETA isn't just getting animals every other shelter is denying.

Second, for their operational costs. If they can't run a shelter, then they shouldn't run the shelter. Period. The shelters I know run on a shoestring budget and manage well enough. Perhaps if PETA were primarily a shelter and not primarily an activist organization, fewer animals would have to die.

Look at PETA's expenses. They bring in $82MM, and spend less than $23MM of it on "Research, Investigations, and Rescue" combined. You're right, the operational costs are too high, and they don't want to cut into their $33+16=$49MM investment in advocacy.

Many people involved in animal rescue rightly criticize PETA for fighting for "animal rights" at the expense of the actual animals. That is why.

Animals are not ours to experiment on, eat, wear, use for entertainment, or abuse in any way. Explore this section to learn more about the issues. - PETA's titular stance on the issues

That is NOT the value statement of an ethical shelter. That is the problem. They even think of having pets as a necessary evil in many ways.