I would never tell any student to struggle financially and/or take on debt to get an engineering degree from Vandy or Rice or WashU (or Northeastern, BU, Lehigh, etc) over Rutgers.
But at 10K/year for Vandy vs. 23K/year for Rutgers, well… that’s an amazing deal for a very well-resourced school. It’s not the be-all and end-all for engineering, but well-endowed private U’s can offer a lot of support to lower-income students that the large publics can’t. We don’t have to put them on a pedestal because of rank, but there’s also no need to pretend that there aren’t advantages to such a school, even at cost parity, and even more so when it’s the best financial deal available. Personally I’d choose Rice over Vandy for MechE (Oshman Design Kitchen and all of the associated project-based learning - terrific program) but regardless, I don’t think the OP is so far out of the running for such schools that they shouldn’t be considered, so long as he can take his shot without getting too emotionally invested.
UAH is great for the right student (I suggest it pretty often as well), but even with the auto-merit, it’s not that much cheaper than Rutgers for this student. You already lobbied for it up-thread, and OP didn’t seem interested. Some students would rather attend their own highly-regarded in-state flagship than go bargain-hunting in Alabama to save <5K/year. Both paths are fine and we need to respect the student’s preferences.