Hi, you're going to post vids, it would be good to edit them to the relevant bits please.
So this is a new horse for you? I'm guessing you're pretty new to horses too? Have you got an instructor handy, or at least other experienced horsepeople you could ask for help from?
He is either not well trained to lead/yield, &/or he knows you're going to let him get away with that behaviour. So no, I would not be riding this horse until he & you understand the basics on the ground first. I also wouldn't bother about a bit, at least until I was sure he understood how to yield and you understood not to drag on the lead. I would however go to a rope halter - he'll be less motivated to lean on it - and a longer rope though. You need to be consistent with whatever 'manners' you want & don't allow him to get away with pulling you around.
But he can't learn that really if you're hanging onto the rope tightly all the time too. You need to give him slack whenever you're not specifically asking something of him, whenever he's doing Right. Because otherwise it's just nagging & he is learning to ignore you, become desensitised to the pressure. Also I think it would help if you didn't keep hassling him to move all the time for the hell of it.
I'd probably teach him the basics in the paddock and wouldn't take him out until he was responsive. I'd be first teaching him to 'respect' my space & not invade it. I tend to use the 'flailing arms' approach - just get big at swatting flies, swinging my arms around randomly, so the horse learns to stay out of the way, or *he'll get himself hit*. This is understood better & no offense taken compared to hitting out AT the horse.
I'd be ensuring he understood how to yield to direct pressure in a variety of ways - eg. Fingertip/halter pressure on his nose to back up, fingertips on his cheek/neck/shoulder & halter/lead pressure to yield his forehand over, fingertips on his flank or rump, & lead pressure towards his rump to yield HQ over. Then I'd teach him to respond to implied pressure - eg. A swung rope or whip, pointed finger, bodylanguage, etc.
Once he was good at that, then time to take him out safely. While I'd endeavor to teach him in as non-confrontational way as possible, especially because he's been allowed to drag you around, it may take a few 'heavy' cues to teach him that the rules have changed. Don't blame him, but do get 'big' enough to be effective. As he's learned to just pull against steady pressure, I think you will probably be best 'correcting' with some 'short, sharp shocks'. Eg if he begins to get ahead of you, drag on you, rather than going with it & continuing to drag, you can give a few sharp jerks on the lead, or use the tail of your long rope to smack his rump & get him to turn & face you. Be consistent with this in NEVER letting him walk over you, and ensure that whatever correction you use is strong enough to be effective, and you'll find you won't need many corrections.
And don't forget to reward him for doing the Right Things too!