The bill passed on May 30, and was signed by Gov. DeWine on June 2. 90 days from June 2 is August 31, so it will be in effect on September 1. Not to mention that Democrats could file the nomination well ahead of September 1, so that it would be on file by the deadline once the new deadline takes effect.
But that tight timeline is probably why the Democrats are planning on holding a remote vote in advance of the convention.
And - none of that addresses the point I made that the language in the OP (changing an existing candidate) isn't applicable when no candidate is yet on the ballot, which should have been a hint not to take anything from the source quoted at face value.
Pointing to Ohio was just a reminder that we resolved all of the issues of filing deadlines that fell before the convention a long time ago, with Ohio being the last to get resolved. If the party resolved all of the barriers to naming a candidate at the primary, they are still resolved by the same means (whether that candidate ultimately named is Biden or the man in the moon). The earlier resolutions were generic to major party nominees (Republican or Democratic) - and were necessary precisely because there is no guarantee that Biden is the Democratic candidate until after the convention (or earlier remote vote by the delegates).