If you are looking for more time around aircraft and actually get time in the cockpit (but not be the sole person in control), becoming a Navy flight surgeon would suite you better.
Air force flight surgeons have a 5-6 week course on how to do aviation occupational medicine, it doesn't have much to do with actually flying. Your job is to work in the clinic 5 days a week doing physicals/sick call for air force pilots.
Navy flight surgeons are trained in a 6-8 month course, 2-3 of those months you spend in flight school alongside normal Navy pilots. The expectations, however, are much less. There's a 5 week ground school component where you are expected to pass the basic ground school tests with the same standards as pilots (covering basic weather, flight rules, engines), but theres so much study material out there that it is not difficult. After that, you head off to primary in the T-6 Texan II where you actually get to fly with an instructor pilot several times. You set the expectations, if you just want to go along for the ride and let the instructor do all the work, they are totally fine with that. If you want to try taking the controls to do loops/aerobatics, the instructor will follow along on the controls and make sure you don't crash it but you'll get to do it. You also get to fly in the TH-57 helicopter with an instructor pilot several times with the same type of expectations.
Once you head out to the fleet, you are supposed to get a minimum of 4 hours of flight time per month (up front at the controls with a designated pilot running the show next to you). Again, if you just want to go along for the ride, the designated pilot will be cool with that but if you want to do more, you can.
As a Navy flight surgeon, You are a class II pilot which means you are never allowed to fly a military aircraft by yourself, there will always be a designated class I pilot at the controls too.
As for hours, you work 2.5 days a week in clinic and the other 2.5 days you spend with your squadron doing admin stuff/flying. Its a pretty sweet gig that gets you actively involved in aviation while also practicing medicine.
The best route to become a Navy flight surgeon: HPSP -> 1 year military internship -> 1 year flight school in Pensacola, FL -> 3 years flight surgeon assigned to either a Navy or Marine squadron -> done with military commitment -> off to civilian residency. You'll get paid about 30k/year on HPSP, 80k for internship year, 100k/year from flight school year through time as flight surgeon. If you want to stay a flight surgeon for your career, you can apply for the Navy Residency in Aerospace Medicine after your flight surgery tour and be board certified in occupational medicine for aviators. You'll then work at a higher up level in the Navy aviation environment (for example, head flight surgeon for a whole aircraft wing with baby flight surgeons working beneath you).
Can't speak for the air force path as I am a Navy flight surgeon, but the Navy route is awesome and totally worth it for someone wanting exposure to military aviation in my opinion.