But is it really a "top priority list" problem, given all the other significant problems the US is facing?

That question isn't rhetorical. I think it's worth examining the demonstrable negative impact of illegal immigration objectively and empirically. Just how disproportionate to their numbers is the harm caused by these people, and what are the trends in terms of the nature of that harm?

There have been some studies done, and it is probably worth it to don my professional's cap and have a closer look. At the very least it might indicate that further, deeper research is called for.