I'll also add that sportswashing doesn't have to be unprofitable for the owner either.
If these numbers are remotely true, then it means that when Boehly paid 4.25 billion pounds, RA made a killing… even though Chelsea was always having to get bailed out by his "loans," his total amount spent on them was roughly 2.6 Billion between purchase, additional investment, and "loans" so he made a killing… even though they were always in the red, and always had to take money from him.
https://www.sportingnews.com/us/amp/soccer/news/how-much-did-roman-abramovich-pay-chelsea/rarsgqv7fhomkdpn5prhdxyhSo it can, yes be a rich guys plaything and yes make some money (eventually) - but those factors are independent of the fact that by virtue of getting economically involved, it builds goodwill. There's a quote about no such thing as bad press - if these sovereigns are getting exposure in the West, it makes them more familiar and that can pay political dividends down the road.