I used to use a heat gun (one of the worst ways to do any hot air, as it was 2kW, without temperature regulation) for getting parts of larger boards - think monitor/TV and main PCB's purely because I didn't want to scrap a lot of consumer electronics, without at least collecting a biscuit tin of bits. I have probably used about 5% of what I got, so arguably not worth it.
With that method, you get a lot of parts off, TH and SMD alike but... given the sheer power and widely varying temperature, by the time you've got most of the board up to the point where solder melts, some parts will be so hot the plastic of connectors melts, IC's become damaged etc... Crude, somewhat fun (yeah I'm like that) but not particularly helpful. It's a case of trial and error, medium setting, and play with the distance.
If space isn't a problem, then I would leave them on the boards, and remove then as and when you need the parts... perhaps just removing "large" parts that make storage awkward, like massive heatsinks Components are safer mounted on the board, and you don't spend a lot of time desoldering stuff you won't ever use.
I'll also add to the "its not really worth it", especially for passives. With caps and resistors sometimes literally 10 a penny, and logic IC's still stupid cheap, it's not something one does to save money. Very few IC's have any resale value, and those that do, are usually very vintage and can be easily damaged when removing. Also the ones that are most prototype friendly - through hole - can be hard to desolder in bulk, because of friction of the pins in the holes.
With allll that said... I'll confess I did spend a few hours years ago stripping damaged/scrap laptop motherboards, and ended up with some small plastic boxes of SMD caps, logic, PSU IC's, and MOSFETs and regularly use those parts for my own projects, where I can deal with the possibility the part is damaged. I only cost me a few hours, no money, and takes up about 30x15x10cm space total for storage. Cut up, levelled and cleaned the bare PCB's to make coasters.