I will boil down EXACTLY what I am saying so you understand it clearly.
EVERY single classification of Japanese verbs contains verbs that end in る. So it makes no sense to group a verb into the る naming as there is not a set of rules that fit them all.
This is the argument I am making in the video. I welcome a video made by you explaining why classifying things as る verbs helps the student. You said that the る drops in every conjugation and that is why they are called る verbs.
However I would like to point out that there are easy examples such as 帰る、走る、and しゃべる that don't drop their る but instead change them to り and ら etc. I can give you verbs in EVERY verb type that end with る. Please point out anything factually wrong in my video. Verbs ending in RU can be of all classes..... EVERY OTHER verb can not. う、く、す、つ、ぬ、ぶ、む endings will ALWAYS follow the same set of rules for conjugations. IF る ending verbs could do the same then 100% I would agree there is a SUCH a thing as る verbs.
These are all る verbs right? Please unify them into ONE rule conjugation then we can talk about how there are る verbs.