Amharas claim it because its Habesha history, the whole Habesha Ethio-Semitic mix started in that area where Sabeans settled
I think you're confused, Amharas arent some kind of cushites conquered by Tigrinya peoples, theyre about 70% cushite just like Tigrayans, only reason some of them go below that is because of oromo invasions and slavery
I don’t think Dʿmt was ever a Sabaean colony, though it was certainly influenced by Sabaean culture, and the influence may have gone both ways. The Arabian genetic presence in the region appears to have been primarily paternal in nature. For example, haplogroup J1 reaches its highest frequency among the Amhara, where it peaks at around 40%, compared to less than 15% among the Tigre and Tigrinya populations, despite the latter living closer to the Red Sea.
The kingdom was also influenced by the Greek settlements north of it. The D'mt elites made Greek the state language; its successor Akum went as far as intervening on behalf of Rome and invading Southern Arabia with Roman ships. Rome provided logistical and naval support. Aksumite ships, sometimes under Roman direction, were able to cross the Bab el-Mandeb Strait and project force into Yemen, Rome already had a failed campaign and keeping a garrison there especially with how nibble the Arabian cavalry were, it wasn't worth it for them, so they used Akum to project power, in exchange for this, they supported them in trade.
