A Constitutional Amendment would only invite the Dems/Leftists to find workarounds, ignore the Amendment (as they ignore the Law), and further ignore court rulings that bolster the COnstitution and the AMendments thereto.
Then why didn't they do that with Heller, or with Dobbs, or with the Harvard case? We can't be defeatist and say that it's no use trying anything because they'll just go around it anyway.
The problem with current policy is that it vests too much discretion in presidential enforcement. There are things to include that can make it pretty tight, including giving states standing to challenge federal immigration policy and the right to prevent illegal immigration on their own borders. Expressly overrule Phyler v. Doe And specify that the constitution shall not be construed so as to require either states or the federal government to provide any benefits of any kind, including healthcare and education, to people who entered the country illegally or are awaiting a determination on asylum claims.
Lots of other things we could do to make it much less desirable to try to enter the US illegally.