British people wanting to live and work in Europe will have to apply under Europe's rules. Seems fair. British people who have been evading existing rules and now have no basis to stay in Europe are being told to leave. Also fair.

Are they politely asked to leave? Or are they thrown into detention centers and humiliated, despite willing to pay for the ticket to leave?

The first one. The second Spanish police throws a British immigrant into a detention centre for staying illegally you will see every tabloid in the UK crying about it.

But yes, British visitors to the EU who cannot enter for whatever reason are just sent back on the same flight or wait at the airport for the next one.

Even if that happens, the people with whom anti-immigration policies are popular and job solicitors in the EU are likely two non-overlapping groups.

There was a genre of Daily Mail comment where Brits who'd immigrated to Spain complained about the number of immigrants in the UK.

Why don't they already? Seems 100% unfair. It's almost impossible for non EU citizens to enter Europe and look for a job if they come from Africa for instance. Why should it be easier for UK citizens?

Because we/they decided it, probably based on the relative economies of different countries, chance of illegal overstaying etc.

I think EU and UK people can visit the USA for a job interview under the visa waiver programme. A Nigerian can't. It's the same difference.

Not all EU people. Some still need visas for the US.