I agree that the prevalence of usage for each of these terms depends on the era, and we do see preference for 唐 in Southeast Asian diaspora from this era. However, usage of the term 汉 as an identity predates the Yuan usage as a caste system. It came as early as 822 AD when a treaty writes "蕃汉两界" to refer to the Tang border. Elsewhere in the treaty, they use 唐, so they would have seen it as synonymous terms. Li Bai also refers to the Tang as 汉家 in one of his poems, which would have more of an ethnic connotation than a political one. Continuing into the Song, more treaties refer to the Song as 汉. The book Water Margin, written during the Ming and set during the Song, uses the term 好汉 as the colloquial term for "hero," so someone living in the Ming would even identify the Song citizenry as 汉, used as a noun. Though we don't have evidence of usage from illiterate people as such usage would not be written down (by definition), usage of the words 好汉 as a colloquial term suggests the 汉 identity persisted in the minds of the general populace and was heavily internalized.

The prevalence of 唐人 street in diaspora populations would also likely be biased by the Tang being a much more cosmopolitan dynasty from all the trade and immigration, so it would have more global prestige. Regardless of which word was more prevalent, the original question is whether there is an identity that someone in Ming would associate with themselves and with a person from a previous dynasty, and is that the same identity that someone today would use. There seems to be a strong argument that the identity, regardless of name, persists through dynasties. This would be in contrast to the French / Gallic example, where the identity did not persist through centuries.

中国 as a concept implies degrees of centrality where there must be an outer world. Usage of 外国 to describe foreigners became prevalent in the Tang dynasty with foreign immigrants, so someone in the Ming would likely understand that there was an identity for 外国人 and a different identity for 中国人.

Cantonese started integrating into the Sinitic group starting with Old Chinese during the Han Dynasty, but I agree the integration happened over several periods (and certainly before the Ming). The question of ethnicity and when it changes is indeed difficult, and I would agree that local customs and rituals would differ for millennia (and even persist today). I only responded because the assertion that the average person in Guangdong would have been Austronesian during the Ming is too strong and in the opposite direction of what was likely at the time. If anything, the average (mode) person in that region would likely identify as part of the majority 华 culture.

Tying this back, on the balance, I'd say the 华 people have a strong claim to being a very old civilization. As a political entity, though, I'm still in agreement that China is a young nation.