Based on my experience (including lack of the ability to stick to any exercise regimen long-term), I second GP's post.
I also agree that doing exercises helps recover that second type of energy. It's a positive feedback loop - but the problem is, you won't be able to start it if you're too depleted on the "starting energy". Like with an ICE car - the engine does recharge your battery, but you won't be able to start it if the battery is already dead.
It's also only anecdata for me, but I do have hobbies that qualify as sports but after an exhausting day at work I want nothing less than to also exhaust my body and this has been a theme forever.
Also doing any kind of exercise in the morning is the worst and it doesn't clear my mind, it makes me want to get back to bed and the motivation to start work is at a complete low. Some people may well feel refreshed or good after exercising, for me it's the complete opposite - but I don't claim authority.
A few arguments, though:
1. to me, it doesn't seem the case of the OP (if it was, they'd be already doing it)
2. I know only few people who consistently run, so my opinion is that the motivation/energy balance is not positive for the vast majority of the people (at least, from the mid-term onwards)
3. regarding the "exercise" concept, it depends on what one means. exercise as functional-training-at-home-or-in-the-park is hard to carry in the mid/long term; it's terribly boring and I believe only highly motivated people can do it. if one talks about organized sport activities, I'm all for it; it's definitely something that I generally suggest, but my observations is that those who are not already doing one in the 30s/40s, don't find sport fun in general and don't stick to (or even start) one.
If you're burned out, you won't have spare energy for anything else, and however you approach it, fitness requires a minimum amount of energy to be sustainable, which you don't have.
Technical solutions may lower the energy threshold, but they won't stick, as they won't lower the energy so much that you will stick to them long term, if you are mentally depleted.
It's true that some people feel a certain rush by starting a fitness plan/activity. But without energy and motivation, in the long (or even mid) term, it will fade.
There are also some motivations (I want to do XYZ), but those are external motivations, not internal; they also don't last.
My advice is to tackle the burnout problem with a therapist, then start a fitness plan, or at least, do it contextually to the mental recovery.