Some martial art (a real one) like muay-thai (no need for fights) works really good for this.

Martial arts are really hard to get into as a beginner. I'm physically fit and fairly flexible, but the dogmatism of many martial arts and the high barrier to entry (extreme strength and flexibility seem to be required even to start) made me really uncomfortable with a wide variety of martial arts. I've tried aikido, capoeira, kung fu muay-thai, boxing — but they all seemed to required beginners to immediately be able to do side-splits, acrobatic flips (I'm in NYC, so maybe the quality of beginners was much higher).

I'd recommend yoga in combination with weightlifting or HIIT training.

It's not a gatekeep. It's just easier to do in group by following the teacher/group.

The "have to be fit to join" is a myth. You can go whatever your fit/fat levels are.

The mindset is "do what you can" and improve step by step.

BJJ could be one to try if you have patience left?

I don't think there's a need to gate-keep exercise.

Start with walking and fast-walking in bursts.