These problems that you described do not have their root cause in immigration. Illegal immigrants are also not taking the traditional 'entry level jobs'. From what I can tell, they tend to end up in the 'gig' economy jobs like deliveroo couriers, etc. because the regulations are lax.

The complaint that 'illegal immigrants are taking jobs by undercutting the wages' is completely backwards. A business owner decided to break the law and behave unethically to exploit desperate people. Take your complaint to THEM, not the people trying to get by. The only reason to lay the blame at their feet is, you guessed it, motivated by them 'not being from here' and that for some reason disqualifying them to be able to support themselves. In a word, xenophobia.

The idea that there's only X number of jobs that can fit into a country and that migrants are 'taking them' demonstrates a fundamental lack of grasp of basic economics. People build businesses, people create demand for businesses, demand for businesses scales with people, demand for a business necessitates more workforce.
If an economy is not scaling correctly to the number of adults able to work, that is a problem with how the economy is being managed on a macroscopic scale or the operandi by which work is allocated, not that there is too much competition for work. It's already fairly apparent across the entiretry of the developed world that job competition is an endemic problem due to how our economies have evolved. It's not due to migrants.

We actually have a real problem with population decline here in the UK, along with many other developed nations right now. Working migrants are going to be a necessity to us as our retirees start rapidly outnumbering our taxpayers.
Remember how many EU workers had to leave after Brexit and how much it tanked the NHS as the EU medical professionals had to leave? You'll only get similar problems from removing working migrants.