They may have a lot of money the next year, but they will spend it as they see fit. It is odd that any scholarship school would have half their team be seniors.
So much depends on your comfort level with this situation and your son's confidence in his ability. Just remember that scholarship players get cut at times, so being a non-scholarship player definitely has some risk involved.
There always are a lot of questions about walking on and preferred walk-ons. A couple important things to think about...
1. The risk is far greater for the player.
2. They may not have any money and/or they don't want to risk it on you.
3. Be careful about getting too big for your britches. I have seen many walk on at big power house programs, accept all the congratulations from friends and family, only to end up crashing down to reality.
4. Remember that the second go around in recruiting is often harder than the first trip.
5. Obviously it is better being a preferred walk-on than the other type walk-on.
6. All coaches are not the same. Some live with their mistakes, most try to correct their mistakes.
All that said, many walk-ons have experienced great success. Sometimes it is the best choice, maybe the only choice in some cases. A warning... Unless you have a written contract, never pay attention to the word guarantee! Pretend that word doesn't exist in baseball. Most coaches end up doing what they feel is best for their program and their career. Over a years time people tend to change their mind about many things. So even though they might have a lot of money next year, they lose half their team, They are forced to rebuild and that means spending their money wisely as they recruit another class. It's difficult for a player to prove his value unless he gets playing time.
Guess my point is it's risky, but so are many things that are worthwhile in life. You can't hit if you don't swing. Best of luck.