This has been a source of confusion for me. I assumed that infinitives were verbs, but that isn't correct.
English grammar seems complicated.
I have tried to avoid splitting infinitives, but sometimes you step all over yourself in the process. In some cases, it is better to split the infinitive.
Infinitives don't act as verbs. Rather, they act as nouns, adjectives, or adverbs
As nouns, they act as subjects of sentences or as direct objects:
To rescue Princess Aurora was Prince Philip's only goal. (subject)
A smart approach is to do a little each day. (direct object)
As adjectives, they modify nouns or pronouns:
His favorite toy to sleep with is his stuffed giraffe. (modifies toy)
As adverbs, infinitives modify verbs:
She ran to stay in shape. (modifies ran)"