Is 'tell to' a modal auxiliary phrase?

No, it's the main verb, followed by the infinitive "to stay."

Then whats the grammatical function of the to-infinitive? And if to is an infinitive marker, why is it left behind in elliptical sentences? I.e "I stayed because she told me to."

The phrase tell me to seems to be a constituent functioning as an auxiliary.

I have the same question about "want to", "have to", "need to", "try to", etc. They all seem to behave the same way.

You have mixed categories there, and these "auxiliary" verbs are classified in two different ways.

1. The quasi-modals or semi-auxiliaries.
Semi-auxiliaries include be about to, be able to, be going to, be likely to, be supposed to, had better, have to, ought to, used to, and would rather. Some are followed by an infinitive; others by a zero infinitive. Like the one-word modals, th

I stayed because she told me to (stay). It is not necessary to repeat the verb. It is understood.

The grammatical function of the infinitive clause is the complement (object) of the verb.

She told me what I must do.

What did she tell you?
She told me to stay.