This is a false dichotomy. While it’s true that the president cannot massively reform immigration policy or adjust the budget unilaterally, they do have a lot of sway in how to handle the situation by setting the priorities of federal agencies and working with aligned NGOs. Instead of restricting immigration, why not work with cooperative state and local governments, as well as NGOs to make resources more available to those who need it? He could also direct agencies like CBP and ICE to stop wasting their money detaining people and ruining lives, and instead focus on helping those not in compliance go through the legal process instead, and provide personnel support to aid those crossing the border using their existing logistical resources.
Just limiting immigration altogether is the lazy approach and helps no one.