Hi,
Ok first things from the video. That is completely normal!
Lovebirds are climbing birds and this is good active behaviour. They use their beaks to test stablity and climb, just like an extra hand! This is normal. Also you need more perches in the cage, those two are good perches as they are natural wood from what I can see but the minimum is 4.
Especially if he's had a foot injury previously you need different perches, so some flat, some round, varying textures and diameters to exercise and support the feet. I would get the foot checked again by an avian vet to see if its healed well, birds adapt around foot injuries but its should be checked. I doubt this has anything to do with the behaviour but please get it at least looked at.
I would also at some point consider a larger cage than that although it will be OK for now providing you let your bird out of the cage to exercise often. Additonally, maybe some more toys would be useful for chewing, the cage is pretty sparse with very little in the way of entertaiment and enrichment. More perches, more toys, foraging items, etc would be good to add. That cage is a little on the small side for a lovebird.
As to biting in the cage. How would you feel if a giant hand came inside your house... pretty scared right? Well that's how your lovebird feels for the moment. There is a guide by Michael on this. In the meantime, use a wooden dowel or perch to take your bird out the cage if he's biting really hard. If he's just nibbling thats fine and normal he's jsut testing your fingers and beaking with you but if he's biting hard this is something you need to address be retraining him. Again see Michael's guides. If a bird bites its not the bird that's doing something wrong, its you. I don't mean that in a bad way like you're a terrible owner because that's NOT the case- we've all had our fair share of bites. Birds work very differently and you may unconsciously be doing something threatening and not realise.

HOWEVER, I don't think he's biting I think he's saying to you "hey where's my food?! when you have your hand here normally there's food! Where is it?!"- he's using his beak on your fingers to search for the food! I think this is the most likely explanation. My cockatiels do this when they are hungry they nibble my fingers for food! It's not hard justa little gentle pressure or sometimes a bit of a nip.
Also his diet? What's it like. I can just see seeds and water? That's ok for now but you need to introduce lots of fresh foods ASAP and preferably some pellets if you can or a lot more fresh stuff which should include pulses, brown rice (cooked) with grains as well as 'table foods' like perhaps a hard boiled egg sometimes or mashed sweet potatoe (no salt, butter but some chopped chilli is good!)
A happy bird is a healthy bird!

and from what I can see your bird is normal and not aggressive and you have done a good job training him so far. Just some slight environemntal and dietarty adjustments and perhaps a little training he'll be great!