"why SNMP is in a home router"
"I don't understand why SNMP is in a home router"
Wtf should it NOT be?
E.g. if every SoHo modem/router had the ADSL Line MIB, there'd be a lot fewer discussions on DSL SNR history etc. My D-Link DSL604+ nearly had it over a decade ago (data was there, but not in the official format). Marginally more recently my BT Voyager 21xx had it. If I knew of a current modem/router with the same capability I'd buy it today (suggestions welcome).
I do agree that having write access to the SNMP stuff in general is rather silly, especially from the WAN side of the router.