I loathe it when people use "Google" as a synonym for "search". It's legitimate when you use to describe conducting a search using particularly Google, but as a synonym for "internet searching" it gives unnecessary focus on a single search engine, one backed by a company with questionable policies.
Once upon a time "Escalator" was a trademarked brand name. Kleenex fought hard to retain it's trademark, and Coca-Cola lost the battle for the word "Cola", keeping only "Coke".
It happens. It's part of language. Over time, former brand names lose their significance and slowly slip into common use. That is a perfectly valid way for words to be coined, and probably is the most common way. That's how language works, and it seems that the word "Google" is going that way too.
You're telling people to not use colloquialisms. I can understand the resistance to an extent, but it seems like not only an effort of glacier punching, but a refusal to accept a linguistic trend that has existed long before Google was doing questionable things.
See also: https://www.gnu.org/philosophy/words-to-avoid.html (and particularly "Photoshop")