Trads have a problem with the position because a permanent married deaconate isn’t traditional-- at least, it wasn’t present in what many trads see as the “golden age” of the Church.
In reality, there were permanent deacons in the Church’s history. And some of those deacons were awesome saints-- St. Lawrence of Rome comes to mind.
But from a cultural standpoint, traditional Catholics view the priesthood with a certain authoritarial reverence (hey, I just made up a word!). And a permanent married deacon smacks of a priest-lite or seems to open the door to married priests. The fear is probably unfounded, but trads are a suspicious lot. They want a clear and obvious distinction. And with due cause, I’d say.
That being said, I think that Fr. Z. of WDTPRS has the best point of view on the subject: Let deacons be deacons! If they have been ordained to that position, let them do their job. Which means they can do their job in the TLM-- who, in the absence of a proper deacon, will just nominate an altar boy to fill that role anyway. If you’ve got an actual deacon there, let him take that role in the Mass.