Hi all,
I haven't used an HG612, but I have some thoughts. I started broadband with a Zyxel Prestige 630-C1 ADSL device that had DSL on one-side and USB on the other, and it did PPPoA.
Looking at the Cisco figure attached from
https://www.ciscopress.com/articles/article.asp?p=363733&seqNum=2 and thinking about the OSI-Reference Model, I think that PPPoE and PPPoA are all layer 2 or layer 2.5 protocols and so more modem like than router like. Routing happens at OSI-RM layer 3 or IP layer in TCP/IP.
The Cisco figure shows how PPPoE could be on a PC, but nowadays we have more features all bundled in the CPE as a home router/gateway, which is doing all the modem & router functions in one box with Ethernet and WiFi on it too.
So much functionality has moved into a single box CPE now that it can get hard to separate functions out.
I got involved in some ETSI technical specifications in the past, and the committee agreed that it was much easier to make the end user devices talk AT & PPP on a serial line when they needed to do data communications (so the device behaved like a modem) than to try to invent a new protocol for bringing up the communications. As PPP had so many supported implementations even in the late 1990s

Cheers,
Tony