Words don't change meaning based on pronunciation. They're different words.

Oh really, what about 'lead' and 'lead'?

They will never be confused because context supplies the clues.

This is often true, but far from universal. Oftentimes there are differences in verb and noun (as with "lead" and "lead"), or transitive vs. intransitive verbs ("seconded"), but occasionally we have problems with two nouns or two verbs with the same form:

"I read that magazine": do I mean regularly, or that I have done in the past?

"The messenger wrote down Lincoln's address": was that his home, or his speech?

The context would need to be quite explicit to work around problems such as this.

If context supplied enough clues, grammar correction software would work much better than it does.

What do you mean?

“Dog” and “cat” have different pronunciations in English, and therefore mean different things.

Dog and Cat are different words.

There are many word pairs in many languages that are written exactly the same way but have different pronunciations and different meanings.

You are thinking of the written language as the fundamental thing. Actually spoken language is fundamental, and writing is an imperfect representation of it.

The verb "lead" and the noun "lead" (i.e., the metal) are two different words, pronounced differently. The fact that they are spelled the same is just an example of how the English writing system doesn't encode the English language perfectly accurately.