Whatever the name, the basic idea is that Congress reserves the power to decide whether the revised bill is better or worse than no bill at all. Under the old line-item veto it did not retain that power.
On a related note, I'd like to see a Constitutional amendment mandating that no delegation of Congress' legislative authority may last more than 30 days beyond the start of the next congress. For a regulatory agency to retain power, its continuance must be approved by 50%+1 of both branches of Congress plus the executive, or by 2/3 of both branches if the executive does not approve. Only 50.0% of either branch, or 1/3+1 and the executive, would be required to kill a regulatory agency.
Just quickly re your suggestion: It doesn't bar congress from enacting broad grants of authority to the executive, which is the problem. I don't think that can be done and sometimes it is wise.
Calling that delegating "legislative authority" is merely rhetorical (though sometimes it does happen to be so).