I think that is the suggestion, yes. Yes, the President has to sign the bill to make it a law, but he signs what the VP tells him to sign.
Seriously, why do you think it is complicated?
I just don't think there's any reason to expect Biden would run his presidency that way.
Aside from rubber-stamping documents behind the scenes, there's a lot of other stuff the VP simply can't do. The SOTU, for one thing.
And I think the Joint Chiefs would raise hell if they were told to get their orders from the VP. Or if their orders from the President had to wait for the VP to weigh in and sign off.
And I think that ultimately, it wouldn't actually reduce President Biden's workload in any meaningful way.
And I think that if it did, everyone would notice, would see it as an abdication of responsibility, and he'd end up impeached and removed from office.
And I think that it's highly unlikely that Biden's sense of himself is such that he'd take a back seat when it came to signing legislation and directing the operations of the federal government.
And I think that "a vote for me is really a vote for whoever my VP is going to be" is an absurd campaign slogan on its face. If the whole point of electing Biden is so that someone other than Biden can be president, why should Biden run at all? Why not just keep his Senate seat, and endorse whoever his VP pick would be, in their own presidential campaign?
After all, the assumption here is that whoever he picked as VP must be pretty damn electable themselves, otherwise, what's the selling point for picking them? "You wouldn't vote for them for president, but if you vote for me you'll get them as president" doesn't make any sense.
And I think this raises serious questions about his capacity as Senator. If he's not up for the presidency, how can he be trusted as a Senator? How much of his recent Senate career is actually being run by his staff without his close attention?
If he lacks the capacity to be president, doesn't he also lack the capacity to be a senator? If he had abstained from running for president, pleading incapacity, wouldn't that be the same as telling his constituents it was time to replace him as Senator, as well?
I don't know where this rumor started, or how seriously people are taking it. But I doubt it's true, and I doubt it would work even if they tried it. And I doubt the people who elected Biden for President would put up with it for a moment if he did try it.