Plus the scope for criminal record checks - again, why would they want to import people of dubious character
FOM is currently in theory limited to those with a clear criminal history. But the problem is enforcement in the absence of meaningful border checks, plus complications where other issues - such as the right to family life - arise.
I can't think of a country which doesn't operate a work visa system that offers both routes linked to a particular employer and more generally. In the UK, Tier 2 work visas are only available to those who get sponsorship by an employer for a particular role, and the employer has to demonstrate that they were unable to fill the post from within the resident labour market. This is broadly analogous to the US L-1 visa. If you lose your job in either the UK or the US, you have 30 days to find another and apply to switch employers, or to leave the country. Obviously the US "at will" employment system is higher risk than the UK, even with the various dilutions of worker protections over the years.
Tier 1 visas are for highly skilled migrants who meet a set of criteria based on age and qualifications, but who don't need a specific job offer (or therefore a sponsor). The US version of these is limited, I think, to those who have a large amount of capital to invest in business that will create jobs.
Tier 3 is currently on hold, but would be the route to open for low skilled migrants post-Brexit (it was designed that way; I don't think that there is a US equivalent). Tier 4 is for students (US has similar). Tier 5 is a mixture of niche elite occupations (eg sportsmen, ballet dancers) and not so elite but in demand for short periods (eg sheep shearers). Again, I'm not sure that the US has a direct equivalent.
The Green Card is a really interesting route, as some people can jump straight into it eg if they have a close family member who is a US citizen, whilst others can move into it having spent time on another visa route. Or certain nationalities (including the Irish, but not the Brits) are eligible to enter a lottery, which is as random as it says on the tin. I've always thought that this is an odd way to run an immigration process, but perhaps no more capricious than some others....