To be clear, I agree with you. But I suspect the Hinch/Harris response would be Flaherty has a track record of several years as a rotation starter and Montero has a couple of months.
Less likely, Flaherty and his agent have told the Tigers that if he's not in the rotation they're going to request a trade (which the Tigers would probably be happy to accommodate given their prospects for the remainder of the season).
If Jahmai Jones is still getting ABs you think Flaherty is suddenly going to be sent to the pen? 🤣 That aside, as many of us have mentioned before, he could be effective in that role given his stuff and that even when he pitches well in a starting role he's limited due to the number of pitches he has to throw to get outs.
In a perfect world I'd like to try Flaherty in a multiple inning middle relief or setup role. Bring him in only at the start of an inning (his tendency to walk guys is less exposed that way) and have him max out for two innings 2-3 times a week.