What prevents it may be that it is not allowed or required. Recall that the essential constitutional issue in Bush v. Gore is the Equal Protection Clause:
Having once granted the right to vote on equal terms, Florida may not,
by later arbitrary and disparate treatment, value one person's vote
over that of another
Ohio has a set of rather specific laws governing absent voter ballots. ORC 3509.05 starts the specification of the ballot return process.
The elector shall mail the identification envelope to the director
from whom it was received in the return envelope, postage prepaid, or
the elector may personally deliver it to the director, or the spouse
of the elector, the father, mother, father-in-law, mother-in-law,
grandfather, grandmother, brother, or sister of the whole or half
blood, or the son, daughter, adopting parent, adopted child,
stepparent, stepchild, uncle, aunt, nephew, or niece of the elector
may deliver it to the director. The return envelope shall be
transmitted to the director in no other manner, except as provided in
section 3509.08 of the Revised Code
The Sec'y of State is responsible with setting up regulations for the implementation of voting, this being the standing rules regarding absent voting. It says (1.04(2))
A voter may deliver the absentee ballot personally or may have a
family member deliver the absentee ballot by the close of polls on
Election Day at the office of the board of elections only. No one may
return a voted absentee ballot to a precinct polling location.
HB 197 was passed at the end of March, which provided emergency exceptions to the ballot return law for the presidential primary (§32 starting 342 of the pdf), but this only applies to the March 17 election: by that emergency provision,
The board shall place a secure receptacle outside the office of the
board for the return of ballots under this section.
In a directive of August 12, the Sec'y of state expanded the scope of the emergency law:
This directive requires the continuing use of that secure receptacle
for the return of ballots and expands its use to include absentee
ballot application forms. The drop box must be monitored 24/7, and at
least one Republican and one Democratic member of the board or board
staff must together retrieve the drop box’s contents at least once
daily. Boards of elections must also retrieve the contents at noon on
October 31, 2020 and 7:30 p.m. on November 3, 2020
Boards of elections must continue to use the drop box that was
installed outside each board of elections pursuant to H.B. 197 for the
2020 Primary Election for the November 3, 2020 election cycle for
the return of absentee ballot applications and ballots.
Beginning September 1, 2020, boards of elections must begin to
provide voters with 24/7 access to the drop box, which will of course,
already be securely monitored. Boards of elections are prohibited from
installing a drop box at any other location other than the board of
There is a separate question whether he has the legal authority to override legislation, but that is up to someone with standing in Ohio to question. (He requested a formal legal opinion from the state AG, but none was rendered).
That does not mean that there is no other imaginable system. But, first, even getting one drop box per county is legally questionable without an act of the legislature; second, increasing the number of boxes without authority is even more questionable; third, leaving it up to the local boards to decide how many such drop boxes there shall be raises the threat of Equal Protection problems of Bush v. Gore. Washington has solved the problem with uniform laws requiring 1 box per 15K registered voters and 1 box per city, town, and census-designated place in the county with a post office, that is, the system is uniform.