It's also one piece of the puzzle. There are other schools that have this type of system in place that haven't been as successful as Vandy. Some of them have. I think the pieces (above and beyond scholarships) are as follows:
1. Corbin. One of the best in the game. I know some here think he's overrated, but I think that's lunacy. There are lots of programs that have been good for decades that have never won a title (MSU, Clemson, FSU, etc.). You can point to individual teams that have underachieved, but also to others that overachieved. Baseball is the hardest sport in some ways to actually win it all.
2. Family aspect. This is part of Corbin's prowess, but it deserves its own bullet point. The players that leave come back, hang around the program, give to the program, and generally invest in it with time and money. When you're a player working out right beside David Price, Sonny Gray, or Dansby Swanson, it makes an impact. Several of the guys actually live in Nashville in the off-season. Dansby Swanson just donated a sound system for the football weight room. The sports support each other in a way that I'm not sure happens on every campus:
https://twitter.com/VandyFootball/status/871860070346260480
3. Baseball is different. The minor league set up helps to benefit academic schools in a way that football and basketball do not. In football or basketball, there are tons of talented players that Vandy can never recruit. So not only can they never help you, but someone who reads at a 6th grade level can actually hurt you when some other school shoehorns them into college and they play against you. In baseball, anyone with serious talent who has no interest in school simply signs with a team rather than go to college. The players who do want to go to college, as a subset, are more academically inclined, and as such Vandy presents an attractive option for them.
4. The socio-economics of baseball as a prep sport. Travel teams cost money, in a way that AAU basketball isn't really comparable. The money and time required by serious travel ball tends to lend itself to higher income families, and higher income families are more likely to want to send their kids to a school like Vandy.
5. The school itself. Academic reputation and value of degree.
6. The city. Nashville has exploded with growth and is generally a very attractive city right now, particularly for young people.
7. Vandy isn't a football school. The lack of success on the gridiron has allowed baseball to carve out its own niche, not in the shadow of other programs on campus.People like a winner, so the donor class can gravitate towards baseball in a way where other SEC schools funnel those donors towards football or basketball.
8. SEC network. Baseball isn't a revenue sport at 99% of places. Which means that baseball is, to some extent, dependent on cast-off from football and basketball. Vandy's still at a disadvantage there, because the overall AD is not a cash cow like others in the SEC. But the SEC network has given a large influx of discretionary cash, and baseball I'm sure sees some of that.