Since you lease the horse for riding only, and he is in a barn that only has one other horse in there so chance of the infection spreading outside of the barn should be minimal, so I think if I was you I would simply take some days off riding and check back in a few days to see if the horse has recovered.
Don't risk carrying the infection to other horses by trying to treat it when the barn owner has said it is fine.
It is very likely the barn owner knows it is in fact strangles, knows it has ruptured and is draining, and for whatever reason has chosen to let it run its own course. While most of us on here will disagree with that, neither you. Or us have any pull when it comes to medical treatment of someone elses animal.
If you are leasing and not responsible for vet care, stick to that, whether good intention or bad, one should never meddle in another horses care without a sound diagnosis by a vet and permission to treat from the owner.
Whatever this horse has (I also believe strangles) his barn owner is aware of and has choosen not to share any information with you other than the horse will be fine. I think you should just leave it at that as attempting to treat something you know nothing about and have been told is fine, will only cause an irritated barn owner.
I agree. I would not treat the horse in any way or even take its temperature. This is not my horse. But if I am paying to lease a horse and the horse is sick and left untreated, what about my lease? I cannot use the horse, and my understanding is that strangles infection can linger for a long time. It has respiratory ramifications, so I assume riding is not a good idea until the infection has run its course.
I go to another barn one day a week, and now I have to worry about sanitizing/bleaching my boots after I visit my leased horse, even though I have no idea if that's necessary, since strangles has not been verified. It's frustrating for me because I just started leasing this horse and now she is basically off limits to me, and may or may not be shedding a highly infectious disease.
My understanding is that following a strangles infection in your barn you're supposed to sanitize (or even burn/trash) everything the horse came into contact with (brushes, halters, lead ropes, stall, tack, etc.). It sounds like a pretty huge job. But if strangles is never verified, none of this will be done. Doesn't that mean the infection could linger for some length of time and continue to be carried around on my boots and the boots of anyone else interacting with these horses?