This is a totally backwards logic.
What actually is being agreed here?
Immigrants are net contributors, meaning more cash is available for the health service.
Government refuses to spend that cash, regardless of who gives it to them.
Therefore... who is to blame for the health service being in a mess?
Because, even critical studies, that move beyond economics, recognise that the main 'cost' to public sector services is in translation services etc rather than direct health care. As immigrants tend to be of working age (and often are working for social care, and the health service itself!) then the issue, if anything, is that we've let a load of the buggers go away, which then creates a vicious cycle where those who are left end up unreasonably stressed as their workload increases, meaning they're more prone to time off, and / or leaving the profession, meaning that those who are left end up unreasonably stressed...
My cousin, for example, is now a team of one, covering an area of 1000 square miles plus. She's seriously considering giving it up because it's too much. Building hospitals is the least of her worries, as she'd still be expected to get to them. OK, training nurses will take too long for her, but maybe we should have fought harder to keep those that were in jobs still there... and I'm not even talking Brexit, I'm talking paying proper salaries and, even failing that(!) paying for proper support for the nurses themselves in terms of mental health and wellbeing(!) Maybe if there was a better avenue for her to actually deal with the stresses she's under, she'd have more likelihood of staying.
We can start with the basics, then look at the grand projects, but whatever we do needs the cash to be spent, rather than denied through ideological reasons. In the meantime, throwing out tangents helps nobody, as the only solution to immigration is no immigration... and then we have even less cash available to fix the problems!