Actually, I think the OP has an interesting point about being asked to go first-name basis when not ready for it.
There's such a thing as insisting on too much familiarity. Insisting that you be called "Mary" not "Ms Smith" is meant to imply that hey, you're an informal and friendly sort. But it also implies the establishment of a connection. "You MUST call me Mary!" implies that you are already familiar, even friends. This is not something one should be demanding, even if you are willing to offer it.
Of course, in today's society, many people are going first-name with everyone - "Hi, I'm Jill your waiter!" "Great, I'm Sam, your customer!". But particularly for those from an older generation, insisting that they refer to you by an informal address may be forcing them to assume an intimacy that they don't really feel ready for. It's sort of like insisting on giving someone you just met a great big hug in greeting. Some people really don't want this much intimacy before they are willing to reciprocate. As much as one may want to be informal, I think one should have some tolerance for people who don't want to be so close so soon. Particularly if you do not intend for your colleagues or customers to have the same privileges as your friends would.
In addition, I do think that in business you cannot, in politeness, expect to be called Ms. Smith or Dr. Jones, if you are calling your customers or clients John and Mary. Any doctor who would be offended by someone saying, "Hello, Leslie, today I'd like you to check out my sinuses," should be greeting his or her patient with "Hello, Ms (or Mr) Franklin".