How can Ohio keep the Democratic Party nominee and incumbent off the ballot?
Every state has its own byzantine maze of ballot qualification statutes. If they didn't, the ballot would be 50 pages long. Some make certain exceptions for "major party" candidates but even those tend to have timing requirements to allow clerks sufficient time to print ballots, mail them to troops overseas, etc. The real issue is that partisan operatives like LaRose are now exploiting every possible area of election law.
Regardless of what Trump says, he appoints exclusively hard-right anti-abortion judges. When you "leave something to the states," assuming he would even do that, the matter doesn't end there. Any state law permitting abortion will be subsequently attacked by rightwing recreational litigators. And if Trump is elected, a greater number of the people deciding those lawsuits will be far-right partisan operatives looking for any reason to make abortion illegal.