He could do that under present law,

And it's unlikely to have the slightest effect. Trump was perhaps the most vocally, performatively anti-immigrant President we've ever had - and he experienced a tsunami of border-crossers seeking asylum in the last pre-pandemic year of his term. They're fleeing what's in their home countries - they're not afraid of a lack of housing or services here.

What about immigration that prevents the building of the wall?

As you point out, the President can't do it unilaterally. It requires Congressional funding. And even if the funding were available, it would take years and years to get the permits (those NEPA reviews are brutal) for the few areas you could physically build barriers. And in most places, you can't build those barriers right up against the border - either because the border's along the river (and thus the barriers can't be built within the floodplain of the river for a host of reasons, including treaties with Mexico) or because the border is in some other type of protected lands.