How convenient for you. The reason why you believe UHC is a useful stepping stone is probably because making UHC a reality will pull UBI within the Overton Window. But perhaps there are other ways to improve UBI's legitimacy. Like... discussing ramifications. Or performing an experiment.
Allow me to reiterate: "Voters' power over a coalition varies directly with the voters' volatility." Once a vote is pledged, the voter's negotiating leverage is lost. It's like haggling. If a potential buyer says "Oh what a nice sweater", the sweater's price is mysteriously doubles. People learn not to say things like that, because they don't want to waste their money.
Agglomeration is one tactic at one's disposal. Walking away is another. Maybe that sounds weird. Because as far as you know, the only force at work in politics is economies of scale. "Join us! Resistance is futile!" True, the power of a coalition (over other coalitions) is proportional to its size. But it comes at the cost of the voters' power over its residing coalition. By analogy. A heavier motorcycle is more powerful. But a lighter motorcycle has better handling. Meditation: why do states/provinces/municipalities exist if the federal government has stronger economies of scale?
UBI proponents should pledge their allegiance to liberals? Wouldn't that be convenient for you. Oops, the price just doubled. Question: how does a store convince a potential customer to part with their money? The store courts them with something of value. And how does a liberal convince a UBI proponent to part with their vote? "Your idea is dumb but mine is better. So buy into mine instead. Whoah, no need to compare -- trust me I'm a professional. Now shut up and give me your vote. By the way, your vote is worthless so you should totally give it to me." The pitch didn't start out bad, but things get sketchy fast. You're essentially trying to conscript (rather than entice) mercenaries into your coalition when you're not in a position to do so.