Limitations
The biggest limitation on this is that then the President is stuck with someone from the other party as a cabinet secretary. While cabinet secretaries are not all powerful, they do have influence over their department.
Another limitation is that it requires someone to be willing to give up their Senate (or House) seat, knowing the impact that it would have on the Senate.
Replacements are appointed by the government of their state. So for this to work, the Senator has to be from a state where the appointer is from the correct party. This is usually the governor, but it's a state process, controlled by state law.
Of course, this only works with Senators that you want to have replaced. So there are many states that might appoint the right replacement but where the Senators don't actually require replacement. For example, Donald Trump appointed Jeff Sessions, who was replaced by Luther Strange. But Trump didn't need a replacement for Sessions.
Many Senators that Trump would want to replace are from states where the replacement would be chosen by a Democrat. The intersection of the two criteria limits the potential candidates. For example, Manchin would have been replaced by a Democrat previously. It's only Justice's party switch that makes this useful there. Again, this would be true for other presidents, not just Trump.
It's also rare for it to matter. Many times the Senate balance isn't so close that switching a single Senator can sway it. And of course, the Senators most likely to switch are those who are most likely to cross the aisle. For that matter, the balance currently isn't that close. The Republicans already have a majority and aren't close to a sixty vote super majority. So this would only matter on the health care vote and other votes that cleaved the same way.
There is a bit of a trust issue. Technically, members of Congress have to resign prior to becoming cabinet secretary. So a president could nominate someone and cancel the appointment after the resignation but before the person took office. Or fire the cabinet secretary.
Previous attempts
It was never described that way, but Barack Obama's nomination of Judd Gregg would have had this effect. At the time, the Democrats had 58 Senators (including Bernie Sanders and Joe Lieberman as independents). Gregg's replacement would have been the 59th and Al Franken the 60th. As it happened, it didn't matter, as they convinced Arlen Specter to switch instead. That gave them 60 votes until Scott Brown took office.
Of course, with Gregg's replacement, they might have had the sixty vote super majority sooner and longer. Gregg ended up withdrawing his name from consideration, citing irreconcilable policy differences.
It's relatively rare for Presidents to appoint cabinet secretaries from the other party. Certainly not unknown, but a small portion of the overall appointments. An even smaller number of those are from the Senate or the House.
Bob Casey was appointed to a position in 1976 by Gerald Ford. This might have brought the Democrats in the House of Representatives below the two thirds super majority threshold on some votes. Or maybe not.
