Correct. We have "été" (summer) where the final "e" has an accent. Also "musée" (museum) where the accented "e" is at the end of the word and not the last letter but still gets and accent. So all cases exists and I don't believe the rule mentioned is correct, especially because it says "a *vowel* gets an accent". Few Is and Us get an accent in French regardless of where they are placed in the word. Maybe they meant "the 'e' gets an accent." In any case, even for the "e" that doesn't sound correct as you demonstrate.
Most accents circonflexes (ex: fenêtre) replace a lost letter after the vowel. In Old French it was "fenestre" without the accent circonflexe.
As far as the "e" I am afraid that there might be no rule as to when it takes an accent and when it doesn't. You should go by the sound. Of course for that, you should know in the first place how the French pronounce it. It's one of those cases, I fear, where you just have to memorize each word--like we memorize the gender of the nouns.
Now if the last "e" is mute/muet it will not get an accent. If it did it wouldn't be mute anymore. In "être" the last "e" is mute because it doesn't get an accent. In "été" the last "e" gets an accent and therefore is not mute.
NB: When you have "er", "et", etc. within one syllable the sound of the "e" is "é". Those are called "digraphs" (2 letters that form 1 sound). Like in "secret". And in this case the "e" is not considered mute even though it doesn't get an accent since it sounds like an accented "e".
Pronunciation is rarely simple (smiles).
I hope this helps.