I always took the the word "tolerance" to mean to endure something until it
I'm a bit puzzled by your example; I'll get to that. But first, let's check a dictionary. Here are the relevant meanings from Webster:
"To allow to be or be done without hindrance, prohibition or contradiction"
"To put up with"
You're certainly not obliged to tolerate your stale house guests in either of these senses. If you ask them politely to make other arrangements, you're well within your rights. But not all cases are like that. Sometimes I'm obliged to tolerate certain things even if it causes me pain to do so. If I don't like it that members of a certain group frequent my favorite coffee house, that's tough. I shouldn't do anything to hinder them, even if that makes my latte-sipping less pleasant. I may also not like the views you express as you address the town council. But I should tolerate your expressing them - whether or not that makes me happy.
The point so far, then, is that there are things we really should tolerate whether or not it somehow makes us suffer. There's room to argue about the cases, but the principle is surely correct. And in those cases, it's not manipulation to appeal to tolerance; it's pointing out what's called for.
As for your example, you say that "our Governments" demand "unending 'tolerance'" of "people who do not pay their way." I'm not sure where you're from or what government you have in mind. I'm from the United States, and I don't recognize this as part of the common political rhetoric when it comes to social services. There's talk of the "social safety net," for example and there are services provided by the government for people in various kinds of need. But "tolerance" isn't one of the usual buzzwords for these kinds of cases.
You might say that nonetheless, governments are implicitly asking us to tolerate people not contributing. However, let's be careful. Many people who get various sorts of government help have jobs, and wish they had better ones, but they can't make ends meet. Some people who benefit from government assistance are children too young to work. Making sure that kids are provided for sounds like something a decent society would do. Some people who get government assistance are disabled. It's not their fault that they can't work. I don't recall anyone using the word "tolerance" here, but I'm quite happy to "tolerate" having some of my taxes go to helping these folk.
Does the system get abused? Yes, though just how extensively is a factual question that I'm guessing neither of us is in a good position to answer. And are there people who get government support (or for that matter, support from private charities) who are just plain freeloaders? Let's agree that there are. It may be inevitable, all the same. There's a trade-off between making a system strict enough to avoid abuse and flexible enough to do what needs be done. That means we're indeed being asked to tolerate something. But we're not being asked to condone or turn a blind eye to cheating and freeloading. What we "tolerate" is that our systems for helping people are imperfect. We tolerate this because we think (or many of us do) that an imperfect system is better than none. That kind of toleration is perfectly compatible with trying to make the systems better. And asking people to have that nuanced kind of tolerance doesn't sound like manipulation to me.
Afterpoint: one way of reading your question was to take it as having to do with immigrants, perhaps especially "illegal" immigrants. That would raise a whole set of issues on its own, but one that's worth noting, of course, is whether it's true that on the whole, undocumented immigrants don't pay their way. Given that most of them work, and do many jobs that might not otherwise get done, that would be an assumption in need of a defense.