Spot on.
A Li-ion battery is called an intercalation battery. That means that the ions physically slip in and out of a 2D graphite anode. When that happens the graphite swells 10-20%. When that happens, it starts to develop cracking and material degradation. The use of silicon dopants and other things has helped that somewhat but it still happens.
To minimize battery size, charge is typically maximized. Thus maximum swell, higher oxidative stress on the materials, higher heat due to losses as the battery charges. All negative.
In small, shallow depths of discharge, batteries can go millions of cycles. They will still naturally age, based upon exposure to temperature and other conditions. But keep that voltage, and this state of charge on a narrow band and they will last much longer.
So while the whole carbon friendly claim is a bit much (unless you have some major price difference per kW/kWh during the day), there is logic to doing this.
People are getting their panties in a bunch because they want to, not because there’s a good reason.