An interesting story by Andy McCullough of the Los Angeles Times was published in 2018 that detailed Ohtani's first meeting with the Los Angeles Dodgers. Because this was prior to the implementation of the universal DH rule, Ohtani would only be able to hit on the days that he pitched, defeating the purpose of being a two-way player and eliminating any chance of him signing with a National League team at the time.
For many of the teams that are in contention for the Shohei sweepstakes, many of them already have a crowded rotations and lineups. Obviously, Shohei would have no problem finding a spot but it would inevitably push out two players and create waste on a team's payroll. For example, the Astros have room in the back of their starting rotation but Ohtani's presence as a DH would force Yordan Alvarez into the outfield, something that could create a defensive liability akin to Kyle Schwarber's -21 defensive runs saved this year.
The Mariners honestly haven't had a consistently productive designated hitter since Edgar Martinez hung up his cleats in 2004 and definitely have room in their starting rotation. At worst, the Mariners would cut loose a couple of bad contracts and trade away their existing pitching talent to acquire more prospects to bolster the farm for the future.