This practice is the norm in part because it's simply embarrassing for a leader and a president to bring up a bill and have it fail.
More broadly, though, it's done because party leaders generally want to protect their party members, and particularly their members in swing districts, from casting tough votes unless it's totally necessary.
Voting for an intensely controversial bill like the American Health Care Act is politically painful, particularly for members of Congress in vulnerable districts. These votes can be used against them in attack ads and hurt their reelection campaigns.
Still, if members of Congress take a tough vote and the bill becomes law, at least they've achieved something. The worst outcome for a leader, though, is if you force your members to take a tough vote and then the bill fails anyway. Then you've forced them to stick their necks out for no benefit, and made them more vulnerable to attacks in the next election.
This is exactly what happened in 1993 when House Democrats voted to pass Bill Clinton's "BTU tax" on energy, a bill that died in the Senate. And even Pelosi let it happen once in 2009, when a fair number of Democrats in vulnerable districts voted for a cap-and-trade bill that, again, died in the Senate.
In both cases, many of those Democrats then lost their seats — and their party lost the House — in the next midterm election. ("Getting BTUed" briefly became a fairly common phrase on the Hill.) And both of those bills at least got through the House; casting a vote for an unpopular proposal that ends up failing in the House seems like even more pointless.
Savvy party leaders want to avoid that bad outcome, so they generally try to only hold a tough vote if they're confident they can get the bill across the finish line, at least in their chamber.
Harvard professors David King and Richard Zeckhauser demonstrated this with a clever 2003 paper that shows that on controversial congressional votes, "small victories" are a far more common result than "small defeats." That's in large part because if leaders figure out advance that they're headed to a narrow defeat on a vote, they preempt that outcome by not holding the vote at all (until they can line up more votes, at least).