The relevant bit was quoted earlier. If he had been a news or current affairs reporter, he would have been in breach of their rules by criticising govt policy. But he isn't. The rules specifically give far more leeway to non-news presenters like him, and there are plenty of examples from the recent past of non-news BBC presenters giving their views on a range of political issues. Lineker himself is a long-standing public critic of Brexit.
The BBC has got its own rules wrong, basically, having initially got them right by deciding on no action only later to change their mind after pressure from the government.
I suspect it hasn't so much "got its own rules wrong" as been forced into bending their own rules to achieve the aims of HM Government. Cunt he might be, but I imagine the BBC DG found himself with a fairly uncomfortable choice (which serves him right). He must have known, or been advised, that the decision to suspend Lineker was likely to result in, at the very least, unwelcome public exposure, but I suspect that whoever Braverman uses as an axe, er, person will have made it very clear that, come what may, Lineker needed to be hung out to dry.
I spent some time working for the BBC. As a freelance, I was spared the indoctrination training, but the impression I had from all kinds of people in the organisation was that the BBC's impartiality was a fundamental plank of the whole belief system. And, back in the early 2000s, that didn't look like such a mad aspiration. Granted, a lot of the talent is, just by virtue of the qualities and values of people who work in creative industries, likely to be somewhat more left-inclined than average, and a certain amount of right-wing ranting about the "lefty BBC" was inevitable, but fairly easily laughed off. But now, the "lefty blob" narrative is mainstream in an increasingly detached-from-reality government that is so hell-bent on pursuing their various lost causes that it cannot even afford to recognise the reality of how what they are doing is coming across to Mr & Mrs Average Britain.
At some stage, there will be a tipping point. This could be it. Or it might not be, and something else will come up. But this government is making increasingly frequent unforced errors, and sooner or later (please god let it be sooner), they're going to step into a hole they just can't get out of.