Thank you @Domen for the link. This article seems to answer my question, so I'll try to summarize my understanding of it here for anyone else that stumbles upon this. Warning that I am just a layman so I may very well get things wrong and will definitely oversimplify (but hopefully Cunningham's Law will hold).
The article criticizes a separate review that gave a history of cellular lactate production for using the term "lactic acid" instead of "lactate" and for associating this production with cellular acidosis.
To explain why the discrepancy seems to exist at all, the article points out that "lactic acid" was first discovered before there was an understanding of acid-base chemistry, so presumably they did not have the proper tools to appreciate and observe the difference between lactate and lactic acid at the time.
The article acknowledges that "lactic acid" production is correlated with acidosis, but emphasizes that correlation does not imply causation.
My first confusion was, well, if lactic acid is produced, how does it not cause things to be more acidic?
The article argues that lactic acid is not produced at all. Not that it disassociates immediately, but that it is not produced to begin with. (I'll summarize their explanation, but not just yet)
The article continues by acknowledging that anaerobic metabolism does contribute to acidosis. The crux of the question then is, where does the H+ come from?
They go into some detail on the chemistry involved here that I won't summarize because frankly I don't understand it all that well. But in short, they claim that ATP hydrolysis is what releases the H+. Since ATP is hydrolyzed during earlier steps of glycolysis, this means it is the earlier steps of glycolysis that lead to acidosis.
Interestingly, the article highlights that as pH decreases, glycolysis releases more H+:
Note, as pH decreases, there is increased fractional H+ addition to HPi for H2Pi, which accounts for the increasing fractional H+ release during the production of 1,3-bis-phosphoglycerate
The article also explains that after 3-phosphoglycerate is produced with an unprotonated carboxylic group, every intermediate afterwards also has an unprotonated caroxlylic group. This is how they explain that lactic acid is never produced to begin with it at all; instead lactate is produced:
The first carboxylic functional group intermediate of glycolysis is produced in the sixth reaction where 1,3-bisphosphoglycerate is converted to 3-phosphoglycerate (see Ref.9, Fig. 5, p. R507). This is a phosphate transfer reaction, adding the phosphate to ADP forming ATP, with the co-production of 3-phosphoglycerate having an ionized (unprotonated) carboxylic functional group at carbon-3. This is key to understanding the H+ load of glycolysis and H+ metabolic buffering from lactate production. Each glycolytic intermediate following this reaction remains in an ionic form.There is never a glycolytic production of a carboxylic acid since they are all carboxylic ions.
So in short it appears that lactate is produced and that it acts as a buffer against H+. "Lactic acid" has been the go-to term due to the history of its discovery, but it's actually lactate and it does have meaningfully different implications as the production of lactate does serve to consume H+. The earlier steps of glycolysis do lead to acidosis, presumably through ATP hydrolysis. After 3-phosphoglycerate is produced, every intermediate following is also a carboxylate (as opposed to carboxylic acid).