I am not sure what the recount does. If it only takes a ballot and checks whether the vote is for one or the other candidate, than the recount is mostly useless. If, however, it validates the ballot, then it makes sense. That is, checks if the voter is a registered voter, if the signature is the same as in the file,* if it was sent and arrived on time, etc.
Obama surely knew how to validate votes (when he ran in a primary for a Congressional seat):
https://notthebee.com/article/just-a-reminder-obama-won-his-very-first-election-by-charging-his-opponents-with-election-fraud
In my state, you have to request an absentee ballot. In some states they just send out ballot request forms to everyone on the voter roles. Either way you have to fill it out, sign it and send it in to election central. When the election official receives the ballot request, he checks the signature to the one on file, and if they match, he sends you a ballot. The ballot package contains an outer envelope, an inner or secrecy envelope, and the ballot itself. There is a perforated flap on the outer envelope. That is where you fill out your name and address and sign it. The inner envelope is blank. The ballot itself contains no personal identifying information. After you have filled in the little circles adjacent to your choice for all the offices on the ballot (president, senator, house of representatives, state senator, dog catcher, etc.), you fold it up and put it in the inner envelope, seal it, put the inner envelope in the outer envelope, and either mail the ballot package to the election center or put it in a dropbox. When the election official receives it, he compares the signature on the outer envelope flap to the one on file, notes in the computer your ballot has been received, removes and retains the flap with your personal identifying information, takes the inner envelope out and passes it down the line. From that point forward, your name and your ballot are forever separated. That's because the secrecy of your vote is sacrosanct. The secrecy envelope is opened by a vote counter, who removes and unfolds your ballot, and sends it through a machine that tabulates your votes.
All this "we can't verify the signatures" argument is just smoke. You could re-verify the signatures on the flaps, but if you found any discrepancies, you wouldn't know who that person voted for, so you couldn't adjust the vote tallies. That is a feature not a bug. Trump was also complaining his guys weren't close enough to the vote counters. How close do you have to be to see a guy opening the secrecy envelope, pulling out the ballot, unfolding it and putting it in the tabulating machine. Maybe the vote counter was sticking Trump ballots in his shoe or something. Massive voter fraud.
