Homosexuality is: forbidden (Lev. 18:22), considered an abomination (1 Kin. 14:24), punishment for (Lev. 20:13), unclean (Rom. 1:24, 26, 27). :vomit:
OK, the death penalty was pretty obviously specific to OT Israel. Not relevant to the New Covenant.
Everything else you said is correct but the death penalty for homosexuality seems to be clearly an OT thing. You never see any support for it in the NT.