Shakespeare didn't write in Latin, but he did write Latin. His plays are full of Latin references -- and often jokes that are only funny to other people who took Latin in school. The Merry Wives of Windsor has an entire scene devoted to it, and even today it can be very funny. Monty Python didn't invent that joke -- and they sure knew their Shakespeare.

Shakespeare was less of a Latin showoff than Marlowe or Kyd. In fact he was mocked for his lack of a university education[1]. But he definitely considered Latin useful, throughout his career, and playwrights continued to reference it for many years.

[1] https://artsci.wustl.edu/ampersand/upstart-crow-shakespeares...

I may be misreading but I'm inferring from the text that Latin drills weren't useless, hence the quote marks. Is it not implying that he was getting taught something that would later become relevant?

Yes but it was already relevant, since it was the lingua franca of higher education. It wansn't like Latin only became relevant later.

I'm just confused about who would even suggest Latin was "useless" in the first place. But maybe something is missing in the quote.

There are suggestions in Shakespeare’s work that he resented the repetition and discipline that characterized his Renaissance education. This is relatable, as learning Latin in a rigid and laborious way must have felt less than practical and not especially useful to students then, as it often does to students now.

The author is pointing out that there was no audience or need for the type of linguistic creativity that Shakespeare was only able to achieve as a result of his intimate knowledge of Latin, until the professional theater emerged during Shakespeare’s very lifetime. The drilling and study in Latin proved, in the event, to be practical and useful in a new and unexpected way.