time - Is it true that English has no future tense?

I'm a native English speaker and I consider myself to have a very competent understanding of English grammar.

Recently, I have started believing that there is no future tense in English grammar.

Here are some examples of sentences that I previously believed were future tense, but now believe are either present tense or invalid (broken) English grammar:

I will do my homework tomorrow.

I say that this is present tense. The full (uncontracted) sentence is

I (have the/a) will to do my homework tomorrow.

Strictly, this sentence refers to the writer's will in the present. The usage seems to have been contracted in recent centuries, but "will" remains a noun and this seems to be the only grammatically-valid interpretation of this sentence.

I am going to the bathroom.

I say that this is present tense.

The only valid interpretation of this sentence is that the speaker is currently going to the bathroom (in the present, right now). Whether "going to the bathroom" means traveling to a bathroom or peeing in their pants is open to interpretation, but this does not affect the tense. The tense is determined by "am", so the tense is present.

I am going to school tomorrow.

I say this is invalid English grammar.

This is as invalid as I am going to school yesterday. The tense is present because of "am", and the sentence can not be validly interpreted as it stands.

In examples such as

We are going to Italy in Spring,

this is invalid for the same reason as above: "are" determines that the tense is present, and then a non-present time period is appended. This is as invalid as

We are going to Italy two years ago.

These are just a couple of examples but I have looked at many examples of future tense usage and I do not believe there is a valid future tense in English grammar.

Question: Is it true that English has no future tense?