What you should search for is “part 141 flight school". Part 141 refers to FAA regs. The three that come first to my mind are Embry-Riddle, Flight Safety, & ATP. I'm pretty sure the Wright Brothers graduated Embry-Riddle before venturing out to Kitty Hawk. I remember their ads in the flying mags back in the '60s.
141 schools are structured like universities with ground school, flight training, curriculum milestones, & testing. (In fact, there are several regular universities that have 141 schools.
https://www.globalair.com/directories/Aviation-Colleges-Universities-38.html ) The operation & curriculum at 141 schools are FAA approved. These programs are designed to feed the airlines & corporate aviation. AOPA lists 436 part 141 programs in the US.
https://www.aopa.org/training-and-safety/learn-to-fly/flight-schools. 141 schools are usually GI Bill eligible.
(Just this month, Delta was the first "big" airline to announce that they would "prefer" but no longer "require" a bachelor's degree for second-officer. However, it is unclear how that new rule will play out over a career. Nor is it clear if not having a bachelor's will hinder advancement into the left seat & movement up the Delta seniority list to the largest hardware. But for all practical purposes right now, your son can't avoid getting a bachelors degree & hope to fly big iron.)
Most of us probably trained under Part 61, a more diy process for General Aviation, but fast-tracking for the airlines wasn't our goal. There is also Part 135, which we think of as charter flights & small corporate (6 seats). A lot of rotorcraft fly under part 135.
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Since your son already has an Army affiliation, I should point out that the Army has an (on-again, off-again) enlisted-to-Warrant Officer flight training program for those without a bachelor's. See a recruiter. From time to time the Army resurrects the Flying Sgts program. I was one in Vietnam. But I think they are gone for good.
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a constant theme in every post here on PoA seems to have something to do with medicals. Before embarking on a long & expensive flight training program: obtain a 1st class medical.
There are many programs that will let you progress from one expensive license or certification to the next fully aware that you cannot pass the physical for 1st class. I know your son is a healthy young 18 yr old bull, but you'd be surprised what a med turns up. I had a USNA-bound nephew who suddenly turned diabetic his senior year of hs. A superb athlete & a scholar. It happens. (DUI Arrests, substance use/abuse, & alcohol are frequent deal breakers. Or at least require special investigation & approval).
Also, a career in the top tiers of aviation means having your livelihood put on the line every 6 months for 20 yrs with each medical renewal. Another nephew of mine, 50 yrs old, has been sweating marginal ECGs the last several 1st class renewals.
Hope this helps. good luck.