From Japanese For Busy People Volume 1, page 244-245 (Appendix).
Unfortunately, there is no escape from learning the verb classes. As shown above, the -nai form reflects the two bigger classes as follows:
Regular 1, Group 1, うverbs, godan: the -nai form always ends in -anai.
Regular 2, Group 2, るverbs, ichidan: the -nai form ends in either -enai or -inai.
Your method of changing the vowel is valid. You should remind everyone that the consonant in front of that vowel changes a little: さない becomes します and す, たない becomes ちます and つ. For beginners, referring to the hiragana chart would be helpful to catch these differences.
I already have a system for learning verb conjugations. I memorise the -masu form (because that’s where my teachers started) and the “verb class” (godan, ichidan).
I use the てform rules (as outlined in other posts) to work out てform. The て form rules are very random looking. If you make the effort to use them every time you see a new verb, and read a lot and do your grammar study, they become easier to absorb.
I read the vowel that comes before -masu to work out dictionary form, -nai form, volitional form, potential form. (This is the same as using the hiragana charts I posted earlier. If you look closely at my earlier copy/paste notes, you will see it’s doing the same thing as changing a vowel. You just have to be careful in the さ た lines where the consonant changes.)
I write into my Anki card: dictionary form, -masu form, -nai form and te form to remind me of what they look/sound like. If I struggle to remember any of them, I rely on the rules for backup. As a last resort, I look it up in the dictionary.
After about of a year of learning verbs by this method, I no longer need to explicitly write down dictionary form, -masu form, -nai form, te form. You can give me one of those forms, and I can quickly work out the other forms.
I agree that a simple bunch of rules like yours is the way to go. That’s what I’ve been doing. However, my rules are different.
I don’t expect anyone to use the rules (or look up tables) forever. It is just to deepen our understanding. With practice, we should get the conjugated form without much effort.
TLDR: memorise the -masu form. Memorise godan/ichidan. Memorise the て form rules. Use the vowel in front of -masu to work out dictionary, nai, potential and volitional forms. When learning a new verb, put dictionary form, -masu form, -nai form, te form into your anki cards. Look at the first three forms to see how they are related to one another.