What is the evidence that Biden would secretly like as hardline of an approach on immigration as Republicans?

To clarify, I don't believe he does - or that anyone thinks he does. My point was in the context of that specific discussion - whether Biden is trying to solve the problem posed by very large numbers of migrants entering the U.S. to seek asylum.

There's certainly evidence that he making some efforts to do that. He re-adopted the same administrative rule that Trump tried to get through, trying to re-impose the "First Safe Country" rule as a matter of federal code. It failed, in the same court and under the same arguments that Trump's effort did. He's done other things - like temporarily closing a handful of border crossings under the pretense of staffing issues until Mexico restarted some of their efforts to control flows along Mexico's southern borders. So the fact that Trump is perceived as "trying," and Biden is criticized as not trying at all, is due in part to perception.

There's a ton of other aspects of immigration policy where Biden is absolutely not on the same page as Republicans generally, and Republican hardliners specifically. But in terms of trying to mitigate the problem of more asylees than the system can handle, Biden is pretty overtly trying to take steps to do that. Those steps are simply limited by the many constraints on his ability to act unilaterally in this space.