Whether he should or will is almost beside the point.
The Tories will want to keep him on until they've decided who will lead the party in the next GE. Until then he's a useful fall guy, still liked by many, so they'll hang on to him and both they and he will try to turn things round to reinstate his popularity. Expect a lot of urging from the Party to "move on" from the current shambles of 'partygate'.
My bet is on Truss, she's malleable and will adjust her position for a sniff of power. She will be set up to ultimately fail, like May and Johnson, and even Thatcher. All were popular with the masses, until they weren't. The electorate can be quite fickle when they 'lend' their vote, as I believe many did to the Tories under Johnson.
That's the problem with being 'popular'... it doesn't last. Johnson doesn't appear to have understood this. His apparent contempt for the Monarchy and the people - instituting rules which he and his party chose to ignore has upset a lot of people. Some of course will easily forgive him, others won't.
That image of the Queen sitting alone on the day of Philip's funeral has hit a nerve - even with anti-Monarchists like me. An old lady, unable to have the comfort of close family around her at a time when she most needed them, obeying the rules that her government imposed whilst they partied is a poignant reminder to many others of the deprivation they also suffered, and I don't think Johnson's blunt apology cut it for a lot of people.
For me, it's not so much the parties, it's the stance taken since the revelation. The arrogant assumption that we should all just 'move on', the apparent lack of genuine remorse from the leader of a party which is, supposedly, dedicated to the service of Queen and Country, to me rings of contempt.