So, reading through the holes in your story: you are not a leader of this club, nor a member, nor affiliated with the college at all. And the domain name wasn't actually "in use" but parked.
And the legitimate leadership of the college-affiliated club was able to prove to NameCheap that they had a right to the domain name, as it was (not a right to your account, but a right to their club's name on the Internet). And NameCheap cooperated in turning over control to those with legitimate rights to it, rather than whoever's credit card was on the last payment?
Am I in the ballpark here so far? Perhaps NameCheap did have ways of knowing who the rightful owner was, and who you are not--especially if it was a personal account, not a "college affiliated" or "faculty" account!
In your headline, you call the club leadership "an unverified third party" but the college, and the club, and its leadership are, in fact, a first party to this domain and its transactions, while you are the third party, and you also have no idea what verification steps were taken by NameCheap on behalf of the rightful owners, the college, the leadership, or their personal identities. You have no idea about what they did with that.
It's not your domain, and you're complaining about losing something that was never yours to begin with. So you helped pay for it. That was a mistake. The way you pay for club assets: your club has a treasurer, and your club has a "purse" or club account, and your club writes the checks. You wanna pay for something, make a donation to your club and/or college.
Thankfully, it looks like the mistakes have now been rectified.