Except that is not how it works. I am sure things have changed some in the last couple of years, but both of my parents died in 2005, both were on hospice at the time of death, both had used up all the Medicare they were eligible for, had spent all their money and had to apply for Medicaid, which was approved for my dad posthumously. My mom never benefited from Medicaid.
Medicare only pays for up to 100 days of nursing home care. 100% for the first ten days, 80% after that. If their diagnosis is that they will not get better and rehab will not help, then Medicare is done. Medicare will pay for hospice. For my parents that meant that any medications or services related to their diagnosis would be paid for through hospice, thus by Medicare, which did save them some money.
My mom died within several months of going on hospice. My dad was on hospice for at least eighteen months - the extra attention really perked him up. He even lived five months longer than my mom; I expected him to die very soon after she did, but all the support he got from hospice (and our family) seemed to keep him going. He missed her terribly and passed away on the eve of their 65th wedding anniversary.
Hospice really helped us and I never, ever, got the impression that anyone there wanted him to hurry up and die.
Perhaps not the caregivers (at least not the ones you encountered), but the administrators have likely been thinking about it. Otherwise, they could be forced out of business. That wouldn't be good for anyone.
Hospice organizations need money to operate just like any other.