For standard SSH, most non-EC DH keyexchange (aka classic, integer, modp, or $Z_p$) and all ECDH keyexchange use predefined groups which include the integer g in DH or point G in (X9.63-style) ECDH. The group is defined by the keyexchange method name selected by the KEXINIT (type 20) message exchange.
Methods diffie-hellman-group-exchange-{sha1,sha256} ([KEX]DH_GEX) in RFC4419, and the little-used variant gss-gex-* in RFC4462, are the only standard ones that transmit group parameters, and they are only defined for DH not ECDH.
The draft you link follows this practice, although the Bernstein ECDH method used for X25519 and X448 actually uses only the X-coordinate, not the full point.