The building may or may not hold back some recruits but THAT is not the issue of support (read as funding) for a team. It's not just the HC salary either, it's everything.
When alumni are not engaged or don't seem to be it's not about the invites. Think about it, it's an invite to a fund raiser, why not an invite to just meet and talk. Somewhere the relationship went awry. Why, who, what, where and reasoning who knows but a concerted effort needs to be made if that is important. I know and have seen this first hand and how that can change.
If the admin does not care and I mean from the TOP-DOWN then it's a losing battle. Have seen this as well first hand and how it affects athletics.
Look out there, the RPI story is not that different than other schools that have been in this situation. Some have found the answer, some not, but the answer always seems to be some mix of the proper coach, his staff and funding.