If you can save up the money to hike right after graduation, go for it!
That said, I personally would have been uncomfortable using up most or all of my savings immediately after graduating, not knowing for sure whether I could get a job in time to start supporting myself before my budget ran out after Katahdin. So instead, I finished school, got my masters (consciously choosing a program that would not put me into debt), worked a few years, then quit my job and turned 25 during my 2018 thru-hike. I found this approach to be ideal for me for a few reasons:
1) I had enough savings that I never had to stress about the possibility of being financially forced off trail in the event of unforeseen circumstances (doctor visit, gear replacement costs, flights home in family emergency, etc.)
2) I had already started putting money into retirement accounts from ages 21-24, so even though there was significant opportunity cost associated with taking a year off from "real jobs" at 25, I knew I already had money in the bank working for me. (Not a lot, but still "ahead of the game" for my age.)
3) I had enough experience in my field that I knew I would easily get job offers once I finished the trail.
4) After three years into my career, the thru-hike was a good opportunity to really reflect on whether my life was going in the direction that I wanted. While I think that time for reflection could certainly be valuable for a new grad (my 500-mile Camino hike at 22 was definitely a good time for reflection too!), I felt like it came at a good time for me on the AT.
Of course, this approach wouldn't work for everyone. Some careers are harder to pause once you've started than others, and some people find obligations in their first few years after graduation that could prevent them from thru-hiking -- a mortgage, a relationship, etc. -- but just wanted to share how I was able to thru-hike as a young person without getting anyone else to foot the bill for me, vlogging, or manipulating sprite young lads.