I haven’t seen a credible argument along those lines. The credible arguments I’ve seen are:

• Google Chrome sending user data to its search and ad business - this is the illegal bundling in antitrust law (e.g. to use Chrome, you need to provide data to its search business)
• Google illegally paid for exclusivity as default search engine on browsers - abuse of monopoly status since it knows most users won’t change the default
• Play Store case (and others) - I think this is kind of BS since Android allows third party stores, and I think they only lost because it went to jury; I think their 30% cut is ridiculous, but it’s in-line with similar app stores
• a bunch of other allegations

I didn’t see any related to advertising Chrome across their services, which indicates to me that it’s more a strong case at all, at least not compared to the other allegations. And I’m not sure how it would be anti-competitive unless they prevented other browsers from advertising or messed with who saw advertisements for other browsers.

Maybe the EU has more relevant laws (e.g. they have a requirement for OS vendors to allow use to choose their own browser), idk, but I don’t see how advertising their own products violates antitrust. I do see a lot of other viable antitrust allegations though.