As others have said, there can be a wide variety of what is expected or can be expected in a lease situation.
Have you ever looked on Dreamhorse? I have seen lots of horses for sale OR lease on there. Would you consider riding a horse at another barn if the owner paid for everything? You might be surprised, but there are people who would just be happy if you would ride their horse for them. I have seen that a lot lately. I've met doctors and others who don't have any time for a horse, yet are attached to the idea of having a horse. Some of them are beginners and would just like someone to keep their horse ridden so when they ever have a day off to ride their horse will be ready for them.
If I were leasing out a horse, I personally would rather have the horse where I knew he would be getting good care. But I have seen ads on the internet where people will let you take their horse and use it for a year. Most of those people seem to want the leasee to pay for everything though, because the reason they are leasing the horse out is for lack of funds.
As a two horse owner, I am constantly looking for someone who will ride my other horse out with me. I have high standards for keeping both horses fit, and it's really hard to find the time to exercise two horses all the time. I would almost pay someone to come out with me. I'm sure there are other people like me around where you live. So if you just want to ride I am sure there are opportunities if you know where to look.