By the most strict rules of the English language, can you actually say "the purpose is to claim that…" or "her purpose is to embarrass him…" etc. Colloquially, this and phrases like it (i.e. phrases with a noun linked to an infinitive via a linking verb) are said often, but I wonder if it is technically correct. If it is technically incorrect, please provide a technically correct alternative.
Yes you can.
Grammatically "purpose" can take either an infinitive or a noun. An infinitive is common.
Some dictionaries consider that "purpose" strictly means something’s reason for existence, not just a temporary intention. But others disagree, and usage certainly allows for "purpose" to mean "intention".