Based on population average metrics, you are obese and your body fat % is a clear and present danger to your health. Therefore any responsible system for programming is going to give you its weightloss program first regardless of what the inputs are, until your weight is no longer a risk indicator for serious diseases. Then you could be programmed to start branching out into sport-specific or goal-specific types of training.
I have been as high as 255lbs in my life at ~5'11", which is about the same size as you given height difference, and never "looked fat" either on account of not getting a "chubby face" and my body distributing it evenly up and down my legs and back. Fortunately for me, I accepted just how high my BF% was regardless of looks and got my act together before developing Diabetes and needing new knees. It is good you have done the same.
30 Minute Workout: Time does not tell us anything, honestly. If you do intense circuit training with no rests for 30 minutes, that is a lot of working out. If you do one set of vanity lifts followed by a three minute rest for 30 minutes, you will not accomplish much.
To accomplish as much as possible towards "general fitness" with only 30 minutes to work, two good options are circuits or complexes.
Circuit: Instead of doing N sets of one exercise before moving to the next, you set up a bunch of exercises in a row and do them in a loop. Cuts down on rest time and setup-takedown time and allows you to maintain an elevated heart rate continuously for longer. Biggest problem for you is likely that it's hard to get 4-5+ exercise stations to yourself in a crowded gym, and gyms are crowded this time of year.
Complex: Perform a series of connected movements all with the same barbell or dumbbells without putting them down between exercises. My favorite: take a barbell loaded with a weight I can comfortably overhead press at least 10X, then do
- 6 Romanian Deadlifts, lean forward and do
- 6 Bent Over Rows, then clean it up and
- Front Squat it 6 times, then do
- 6 Overhead Presses.
Doing that as many times as you can in 20 minutes keeps the entire body working and no time lost changing plates/waiting for equipment. After you've done it a few times you'll be able to choose a weight that lets you keep it up for the amount of time desired with reasonable between-set breaks.
NB: These are advanced movements that require practice to perform safely. Please try to find a capable person in real life to teach them to you.
Just out of consideration for your time driving to and from and changing clothes, 30 minutes seems like a short program at a commercial gym!
My goal is to lose weight and get healthier: This is all about what you eat. You can slog on the treadmill for two hours and if you finish it with a 32oz gatorade and a powerbar, or go out in the evening and put away 6 beers, not much progress will be made. Fix your diet and any activity will be plenty to lose weight. Just the team sports would be fine if you're eating properly.