Mitch isn't going to answer the request from the governor, either because he is already dead, or so incapacitated that he cannot communicate. The GOP in Kentucky have long-known that this day would come, and changed the law so as to require a special election, and NOT allow the governor to appoint someone to fill Mitch's term. Covering up Mitch's infirmity or death is vital to the GOP. Rather than acknowledge the fact that he will never again appear on Capitol Hill, they are obfuscating so as to avoid the loss of the seat to a Dem.
As for going himself and checking out Mitch's status, I'll wager the governor has already sent someone, and they couldn't see (or view) Mitch, 'because he is resting'...
Fetterman thinks that when he flips to the GOP that (1) he will be trusted by his new party (2) he will be rewarded with favorable committee assignments and (3) that he will be treated favorably by Der Dotard. It will take about a week before he realizes that none of this will happen ...