Test: if the word fits in the gap after the or a, as in She gave a ___, it is a noun. Speech fits, so it is a noun.
Speech is never a verb in standard English. The verb in this family is speak: They speak clearly, not They speech.
Recognizing speech in use
Example: She gave a speech at the wedding.
Uncountable use: Losing his speech after the stroke was hard for him.
Speech stays a noun even when an adjective comes before it, as in His slurred speech worried the nurse. The describing word modifies the noun; it never turns speech into a verb.
In the pattern make or give a speech, the word speech appears with the verbs make and give, and the article a before speech marks a noun slot, so this pattern is another way to see speech working as a noun. Example: She gave a short speech at the wedding.
Fixed phrases keep the noun pattern even inside longer expressions. A clue is a noun before speech, as in acceptance speech or free speech. Example: Her acceptance speech thanked the whole team.
The look-alike case
The confusion comes from speak, the related verb. They speak clearly is a full verb, so learners expect speech to act the same way. It does not; speak is the verb form, speech the noun, and swapping them changes meaning rather than part of speech.
A more reliable test: a verb can change form to show tense. After a subject, speak becomes speaks, spoke, or speaking, which is exactly what verbs do. They speak clearly is fine, while They speech clearly is impossible, and speech offers no such forms at all, so the verb slot is filled by speak and speech can never go there.
The article slot test handles the countable uses of speech: in make a speech the word sits after a as a noun object. Several guests made speeches after dinner confirms it, and the verb choice made or gave does not change that. The uncountable uses, where speech appears without a or the, do not pass the article test, but they are still confirmed as nouns by the other distinguishing clues in the list.