I'm mostly thinking of the Uxbridge branch and it being shared between the Piccadilly and the Metropolitan line. Untangling that is going to be pretty difficult.
With some more rolling stock it may be possible to extend existing Northfields terminators westwards. But, even if you can physically run more Piccadilly line trains to Heathrow, that doesn't help the situation at intermediate stops. When most passengers are headed for central London, using a stopping service to carry them means doing it slowly and causing capacity constraints for people who don't have any other option.
The point is still that Heathrow is so big and important as a trip generator that you have to think about it as an entity like the City of London or Canary Wharf in terms of transport planning. It's not really that hard to manage transport planning for urban areas like these, as they're made up of a wide range of individual companies and trip generators. Even if there were a major shift away from banking, places like the City and Canary Wharf would just end up hosting the replacement white-collar workers in insurance or audit or software engineering or whatever else.
Meanwhile, Heathrow has precisely one economic purpose: an airport. There are plenty of non-airport businesses around but the vast majority are only there because of proximity to the airport and its operations. So, any transport links to Heathrow basically live and die by the number of passengers and amount of cargo flowing through it. It's kind of like the London commuter rail TOC franchise contracts being based on London city centre employment statistics. If the government did stump up the cash for improved surface transport, then it wouldn't be happy if the private owners/operators didn't match their side of the business case bargain and invest in their own corresponding capacity improvements.
The National Audit Office reclassified Network Rail to be a public body, so that its finances counted as public finances rather than the finances of a private company. Doing so meant that some of the numbers couldn't be fudged any more, but in all reality the financial markets already treated it as such anyway. They aren't dumb. I think Heathrow is in this boat now anyway. Investors and asset managers essentially treat the core Heathrow land and operations business as quasi state controlled, because it is so constrained by government policy anyway. It can't go bust, it can't downsize, it can't expand, it can't change most of its operations without government approval. So, if a government took the step to renationalise those core operations, it would be an economic non-event. So long as it still contracted out day-to-day operation and the various bits that can actually operate privately like airport servicing, it wouldn't make a difference. Private financing for expansion will never happen if there's a real risk that a future government will come in and clip the wings of the aviation industry for environmental or other policy reasons, making it impossible to repay the loans. If, however, the state borrowed the money to build more runways and surface rail links at Heathrow, then the markets would see that as having balanced risk and reward. If the government were then stupid enough to ruin its investment, it can pay the price.