Unfortunately, if you want anyone left of centre to run a country, they need to be a white male, charismatic and from the home counties. Same in America - if you want a democratic president they need to be a white male, charismatic and perferably from a Southern state.
Being from a minority, and also socialist, is too much for most people. It's unfortunate but true. Hence why the Conservatives can field a female or south asian PM without much blow back from the media. And hence why the US swung so wildly to the right following Obama's presidency.
For that sole reason, Rachel Reeves would have been a disaster.
Although I don't know too much about Burnham, he must have some charisma to be able to swing sitting MPs to get rid of their leader while not even being a sitting MP himself. Manchester voters seem to really like him, but whether that translates "down south" is another question. But in his favour, he's white and male.
If he's smart, he'll court Murdoch and his ilk. It's really the only way to get the papers off your back - even in times where the internet rules, it's surprising how much sway the Times and the Sun still have.
That means you can't have a true Labour government - you're basically the Lib Dems in a Labour coat. So be it - I think most people are actually closer to the centre than they'll readily admit. It won't stop the Russian and Chinese bots attempting to destabilise your government, but that will happen whether its Labour or Conservatives. The only party they won't attempt to destabilise is Reform, because they know that they'll do a great job of burning down the UK all by themselves.