My suggestion, since you mentioned this horse doesn't think "stop" even when rein tension is added, I would practice the heck out of one rein stops. Clinton Anderson does this, I'm sure you can find a video of CA one rein stops on youtube. I think it really gets your horse in the right frame of mind for stopping. I'd practice one rein stops at a walk and trot- every time you slide your hand and just before you but tension on that rein, say "whoa".
After a few lessons with this, walk the horse as normal and give the rein some tension, and say "whoa". If he doesn't respond timely, one rein stop him. It might take you awhile, but he'll figure out it is much better to stop at "whoa" or rein contact then a one rein stop. Eventually, you'll be able to stop doing the one rein stop and he should hault on rein tension. Once he's respoding to rein tension, build on that. How I do it is I say "whoa". (Whoa to my horses means they should have stopped yesterday
. I take whoa very seriously). Anywho, if he ignors the whoa command, bring those reins up and stop him, even if you have to go back to the one rein stop. Then immediately
back him up. Backing up is kind of my 'punishment' to my horses for not stopping quicker. I don't rip on their faces or spur them when I back them, I just put a little fire under their hinny and let them know they didn't do it good enough. It is a bit of a process, but if you go small step by small step, I think you'll see results.
Good luck and let us know how you're progressing. Each horse is different, so hopefully some others will have other methods for you to try.