Type "visa sponsorship" into a job board and you will get results. A good number of them are adverts saying the employer will not offer it.
This is not a trick played by job boards. It is what keyword search does. The phrase appears in the advert, so the advert matches. Whether the sentence around it is an offer or a refusal is not something a keyword index has an opinion about.
What the refusals look like
They are rarely a heading. They are one line, usually near the bottom, often inside a friendly paragraph:
"Unfortunately, we cannot offer sponsorship or accept applications from those on a time-limited visa."
That is a real advert, from a real licensed sponsor, and every word a keyword search is looking for is in it. Others read "we are unable to offer visa sponsorship at this time", "applicants must have the right to work in the UK without sponsorship", or simply "no sponsorship available". Same meaning, different words, and a phrase list only catches the ones somebody thought to add to it.
Why the wording varies so much
Because nobody writes job adverts to a standard. The same provider will refuse sponsorship in three different phrasings across three vacancies posted the same week, depending on who wrote them. Some negate with "unable to", some with "cannot", some with "not in a position to", some by requiring existing right to work rather than mentioning sponsorship at all.
Reading these properly means reading the whole sentence and working out what the negation attaches to, not matching a phrase. It is the difference between "we can offer sponsorship" and "we can offer a great team, though unfortunately not sponsorship".
What the numbers look like in practice
Across the live care vacancies we hold, several hundred adverts rule sponsorship out in their own words. Every one of those is an application somebody would otherwise have sent, and a week somebody would otherwise have spent waiting on it.
Removing them is not the clever part. The clever part is not removing the ones that are fine: an advert that mentions sponsorship in passing, or quotes a policy, or says nothing at all. Silence is not refusal. Most care adverts say nothing about sponsorship either way, and those are worth applying to if the employer holds a licence.
What to do with this
If you are searching by hand, read to the bottom of every advert before you apply, and read the sentence rather than scanning for the phrase. If an advert refuses, note the employer: it may still be worth applying to a different vacancy from them later, because the refusal is usually about that role rather than a policy.
And check the employer against the register before any of that. An advert that says nothing, from an employer with a licence, is a better use of an evening than an advert full of the words you searched for.