I know that in their 2000s era of F1 racing, their participation was a sort of carrot to offer their top engineering talent. Instead of a dedicated F1 team, they would rotate other people in from the auto company as a reward for their good work. Given the team was inconsistent, not really a great strategy for winning, which is why they pulled the plug.
I believe they ran the programme with permanent personnel in the McLaren/Red Bull era, but it didn't take long for them to realize that being Red Bull's 'partner' meant 'punching bag'. After having a bad go with McLaren and then RBR, I don't blame them for pulling out.
I'm going to guess that they've got wording in the Aston Martin contract that if anything goes wrong, the team will take the blame in public. "Yes, that's the third power unit that blew up in 2 race weekends. We just don't have the cooling pods to control the world beating engine Honda has given us"