Given how fragile are units overall and with the premise that units "loaded" inside a commander can heal (which is according to this post https://forums.civfanatics.com/threads/little-questions-answers-thread.694802/page-53#post-16799520 of @AriochIV the case as soon as a commander does not move), the Field Medic (? - unsure about English in-game naming) promotion inside logistics might be worth (as I assume the boni apply to loaded units as well). If a unit is heavily wounded and you can't retreat it to a save space outside the battlefield, sneaking into the commander's pouch can be a saver. On the way to this promotion you gain the extra spaces and with the Old Guard promotion you can increase the durability of the commander. All together, I wonder if that "mobile fortress" approach could work out, especially when pairing it with a 2nd commader providing the classic offensive punch.
Yes, units heal inside a commander as long as the commander does not do anything, at least not anything that would spend movement points. And having a logistics commander for that purpose can help. It is still kind of slow in enemy territory, since even with +5 you would need 10 turns of the commander not moving until a near-death unit is healed. It is more useful defensively, especially if you do not have many urban tiles around. Stick your units into a commander and let the commander stand on an urban tile and all the units inside benefit from the extra healing.
Using Old Guard on it is a problem though: If your commander takes damage, the units inside also take damage which is not productive when you want them to heal. So it is always better to stick a healthy unit on the tile with the commander (and if there is no healthy unit left, you probably should pack everyone in the escape cars and get the hell out of there). I feel like I messed up whenever a commander tanks a shot and while Old Guard would soften the blow, I would rather avoid the mistake in the first place.