That’s precisely what a lease is in terms of responsibilities. Yes, there is no valuable / sellable asset at the end of the lease period, but that’s why a lease price (if there is one) is much lower than a purchase price. It’s a legal agreement that essentially confers the rights of ownership to the leader for a specific period of time. The benefit to the leaser is that they have no long-term responsibility to the horse, should it become unusable. The owner bears all the risk
It’s not much different in many ways than leasing vs owning a car. When you own you pay a lot more money upfront but you own the asset and can dispose of it as you wish. You are responsible for all the maintenance and repair costs, fuel, fluids, etc. for as long as you own the car.
When you lease you pay less money, and are subject to some restrictions, such as limitations on mileage. At the end of the lease period you can walk away free and clear. You are responsible for all the maintenance and repair costs, fuel, fluids, etc. for as long as you lease the car.
Are you perhaps mixing up a lease - which is a formal, legal agreement - with an arrangement to have someone look after your horse in your absence? Those are two very different things.
When I went to school overseas for a semester, I arranged for someone else to look after my horse for four months. I continued to pay his board, farrier, and routine vet care. The caretaker was responsible for taking care of him, keeping him fit, and was required to take weekly lessons with my coach at her expense. That was not a lease. I suppose I could have looked for a leaser, but it didn’t seem worth the hassle for such a short time period.
Think of it like asking someone to house sit while you are on vacation. They look after the house, water the plants, mow the lawn, etc. but don’t pay rent.
Leasing your house to someone is quite different. They have legal rights to the property during the term of the lease and are usually responsible for routine costs like electricity, heat, etc. Major repairs such as a roof replacement are usually covered by the owner or by tenant insurance held by the tenant, in the case of accident, fire, theft, etc.