I don't really know of an Air Museum that doesn't have money issues, with the possible exception of Yanks Air Museum. There, if the owner wants a project done, the money becomes available to get it done. It's not that the money isn't important; it's more along the lines of the fact that there is no board of directors. There is one person who can tell you whether the money is there to get a part or get a service done, and he is the person who chooses the projects that get completed. So, you CAN get an answer when you need something. It isn't always "yes," but you CAN get an answer and proceed or not.
I'm sure there are other, privately-owned museums similar to the situation at Yanks. It all boils down to how many people have to agree on whether or not to do something.
By way of example, Yanks recently completed restoration on their F6F-3 Hellcat. It sat for a couple of years in an almost-completed state because the wing tank fuel bladders needed to get replaced. They had to find someone who could make the fuel cells, negotiate the task, send the drawings, wait for them to get made, and wait for them to get shipped. Once they arrived, the guys who had been working on the Hellcat, but were on other tasks at the time, could continue with the Hellcat. They waited for many things, but not the decision to continue when the original fuel cells were declared unairworthy. That makes restoration slightly different from place where a task may go from top priority to back burner if an expensive item suddenly crops up and the decision makers discuss it for several years as funding is identified.
All that really means is that unless someone with money owns the Air Museum, the revenues mostly will not cover ALL the projects that museum directors WANT to get done. So, they have to choose and prioritize where their funds get spent. At Yanks, the projects get prioritized, too, but it isn't by committee and it doesn't take a lot of time to get a yes or no answer. The restoration time is the same either way, but the waiting time for decisions is much less.
At the Planes of Fame, they have considerable experience restoring old aircraft. So, the general costs are pretty well-known and be estimated reasonably accurately. Also, Fighter Rebuilders is there at the museum. If the museum decides to restore an aircraft, and if something is beyond the skills of the volunteer people (not exactly uncommon), they have a choice to get it professionally done by Fighter Rebuilders or even to get someone there who can give advice on how to best proceed.
Sometimes, the paperwork is almost as heavy as the project!