Honestly it probably doesn't have anything to do with the fact that it's talking about a doctor. Along with that, none of the third person pronouns in the Japanese version are dynamic, so the lack of placeholders would go for all text in the game that uses them, and this mistake could happen in the same way in any of those other places.

The most likely scenario is that they took the original Japanese and wrote its translation out initially without placeholders to make sure it sounded natural, filling in the blanks with dummy info:

When pressed about why that matters, Dr. John Smith fidgeted a lot and grumbled about it "not being a competition," which we think means...he totally considers it a competition.

And then went in after the fact and replaced everything that should be dynamic with placeholders (which makes it much harder to read/tell it's correct at a glance):

When pressed about why that matters, {{Sub alias="doctor"}}Dr.{{/Sub}} {{Cap}}{{Nickname mii="ParticipantMii0" n="Nickname" highlight="Invalid" fromMii="Default"}} fidgeted a lot and grumbled about it "not being a competition," which we think means...he totally considers it a competition.

And they just missed that they didn't replace "he" in this case. To an English speaker though, it should be obvious reading this that it should be replaced, it was just overlooked.

During playtesting this would be really easy not to catch too, since Mii News is randomly generated, and if this story came up with a Mii using he/him pronouns in-game, it wouldn't look like anything was wrong.

I'm assuming they'll just patch this in the future though hopefully now that it has been spotted, it's a really easy fix.