Your assumption is wrong. Only finite verbs have tense, not nonfinite verbs. The bold verbs here have tense:
- Singing songs at night was a lot of fun.
- I asked them to sing me a song.
- I made the dog leave us alone.
The bold-italic verbs here have no tense:
- Singing songs at night was a lot of fun.
- I asked them to sing me a song.
- I made the dog leave us alone.
Only finite verbs have tense, number, and person. Gerunds and infinitives have no tense, number, or person. That’s why they are non-finite.
Sentences do not have tense. Only finite verbs do, and you can more than one of those in a sentence.
- I am looking for the person who left me a note.
There you have a perfectly normal sentence with two different finite verbs, one in each of the two English tenses, present tense and past tense.
Since the main verb is in the present tense, I can understand why you would call it a present-tense sentence. But I would be careful. This you cannot classify if you go that route:
- His friends were and are a great help to him.
There you have two verbs in both tenses each sharing the same subject.