You know what that sounds like? It sounds exactly like Woodrow Wilson's strategy for passing the League of Nations Treaty. Barnstorming, refusing to accept amendments, going after Senators who opposed the treaty...that's pretty much what Wilson did.
It doesn't work. Presidents have extremely limited ability to convince Members of Congress to do what they want by going over their heads to constituents. There's some debate about whether they have any such ability, but even the optimists (if you want to call them that) don't really think there's a lot of opportunity. And, in this case, the key Senators (Lieberman, Nelson, Snowe, and Collins) were each pretty safe from such appeals; Nelson is in an anti-reform state, the two Republicans are very popular and their real danger is probably a future conservative primary challenge, and Lieberman...well, we've done that one to death. I can't prove it, but I'm very confident that both barnstorming in his state and threatening his committee would only annoy him more.
A much more plausible strategy would involve finding some substantive trade-offs that marginal Senators might go for. But they've probably tried that; it's likely that that's what worked for Landrieu. And Matt is completely correct:
[P]eople sometimes write about this as if there are 57 Senate Democrats itching to do a health care reconciliation bill, being held back by Barack Obama and Harry Reid. As best I can tell, though, the reason the Senate leadership keeps taking reconciliation “off the table” is that there’s very little support for it among the caucus. For starters, Kent Conrad, who’d be in charge of a reconciliation bill, seems to be against it. For another thing, there are doubtless many Senators who are much more comfortable being one vote out of 60 or 61 for a bill than they are of being one vote out of 50 for a bill that “Republicans and moderate Democrats” oppose.Yup.
Anyway: the Wilson strategy? I'm not saying that it automatically leads to Hitler, but I'd just as soon not try to find out.