H.R. 3364 isn't a particularly great example here, because there wasn't very much time between its introduction and signing, never mind the House vote. In fact, half of the time spent between its introduction and its becoming law was the President considering what to do (sign, veto, take no action).
Actually, I would contend the opposite. Most bills don't have a stack of cosponsors, so as a generalisation, that doesn't hold, and H.R. 3364 is one of the best examples of extremely strong support in recent times, even the Republicans actually put country over party idiocy on that one, and it's rapidity shows the strength of support, not it's cosponsor cheer squad.
HR 676 is an interesting one for several reasons, one, it has 120 co-sponsors or heading towards 60% of Democrats, but that is highly abnormal, two, it's going to hang around for a long time, and to a certain extent they're tying people's colours to the mast on it through cosponsoring it as a show and a straw poll, and thirdly, he very may well support it, but his team 2020/2024/2028 advisers are telling him to sit out
for now, which might be getting closer to your point, but it really doesn't hold as a generalisation, and barely holds as tea leaf reading or entrail divining. HR 676 is a bit of a quagmire, and not a true litmus test yet, no matter how much some people are trying to portray it as one.
You're also suggesting with your comments on the difference between voting and sponsoring that he wouldn't vote for it, because you're speculating (possibly unwisely) that he believes the Obamacare status quo is the superior option (or do you think he doesn't support Obamacare either?), or that he might vote for it, but only because he's going to be whipped into the new party line, which isn't going to happen with the current House, Senate and 'Executive timer', etc. (and SERIOUS debate). No, he isn't leading on this, but no one can lead on everything, and you can very strongly support something you don't cosponsor, and you shouldn't cosponsor anything you haven't read backwards, (or possibly are intimately involved with the negotiations and drafting or vote lobbying and bill presentation) and don't want to be (or have the time to be) dragged into hearings over, etc. You just can't pretend or insinuate people are only for things they cosponsor.
(not to mention absence of evidence isn't evidence of absence.)