It's interesting to compare the accent of native English speakers and native Finnish speakers in Latin. :-)
Finnish has a phonemic orthography, almost. So it's quite easy to just read Latin the way it is written.
Italian has almost phonemic orthography, as well, yet the Italian accent can be quite noticeable.
https://areena.yle.fi/1-1931339
and Latin students worldwide do love it!
Many high school Latin teachers are actually not very proficient in Latin, they are merely capable of teaching from a syllabus that is usually quite modest. Except for some elite schools, Latin is taught just to give students a taste of the language and expand their vocabulary in their native language by recognizing Latin borrowings. The pupils aren’t intended to actually master the language and read non-set texts comfortably, let alone follow Neo-Latin materials or practice spoken Latin. If the students surpass the syllabus, that makes things difficult for the teachers. If a student has taken up the language so enthusiastically, the teacher may simply demand that he/she simply test out of the course instead of continuing to come to class.
You’re not wrong that most Latin classes have somewhat different goals than modern language classes, but that doesn’t necessarily make them less rigorous, as you seem to be suggesting. Deprioritizing speaking, for example, makes sense for s language where students are...unlikely...to encounter a native speaker in the wild. We did eventually get there when doing poetry and other stuff meant to be heard, rather than read.
I also can’t imagine many teachers getting upset or uncomfortable with interested students. All of mine would have been over the moon if they discovered us studying more outside of class. I think this especially true for Latin, since it is such a niche field that people are certainly not in for the money!
Source: 3 years of Latin at a public high school, followed by a linguistics and CS degree.
Nuntii Latini finiti
Nuntii Latini Radiophoniae Finnicae Generalis, qui inde ab anno millesimo nongentesimo undenonagesimo (1989) iam triginta annos septimanatim emittuntur, post hanc emissionem finiuntur et decreto moderatorum radiophonicorum post ferias aestivas non continuabuntur. Auscultatoribus, quorum grex ad omnes orbis continentes amplificatus est, propter fidelitatem gratias quam maximas agimus et valedicimus.
A translation (not idiomatic English, staying closer to the original, but better than Google...):
Latin News Terminated
Finnish General Radio's "Latin News", which have been transmitted from here since 1989 already 30 years weekly, will be terminated after this transmission and, by decree of the radio directors, not continued after the summer break. To the listeners, whose flock has grown to all continents of the earth, on account of their faithfulness we give the greatest possible thanks, and bid farewell.