I would argue that their leaving and setting up a new party forced Labour down the modernising route, as they saw the loss of their broad appeal. Labour took a long time to deal with their extremists, Kinnock and Smith both sweated blood to leave an electable party. I am still not sure Blair would have won if the Tabloids were not so keen to expose Tory corruption. Murdoch switched, Blair and Brown proved to be highly competent, and continued to win, improving the economy, and reducing debt. Whether it was socialism lite or Tory Lite, it was seen by the electorate at the time as fairer, with a broad appeal.
You could also argue that, despite the ideological dominance of the Eton Old Boys and their extreme agenda, the reality through the pandemic has been closer to socialism than anything Labour would have dared. You have pointed out many times furlough is about protecting businesses, but it has been spun as protecting ordinary citizens. The NHS, by necessity, has had the support of which it has been deprived for eons, public workers have been seen by the public as heroes.
I still think the corruption stories have a way to run, at some point it will get too much, and Boris will be sacrificed. There will then be a battle between the two wings of the Tory party for dominance.