Likely! It worked against SEGA, in the Sega v. Accolade case. Unlicensed Accolade (and later, SEGA Dreamcast homebrew games) displayed a Sega copyright logo because they were (believed to be) needed to run arbitrary executables.
As a side note, although Accolade won this case, the injunction that SEGA won earlier prevented Accolade from selling product for a while and that cash flow problem motivated their primary investor to step in and remove the company's founder and all top executives and bring in their own team which directly led to the end of Accolade as a going concern (my opinion). Of course in this case, Amazon isn't as vulnerable as Accolade was back then.
Amazon could try that, but it wouldn't stop there. they'd end up in a constant arms race.
also, that "you know, for search" tagline is featured on t-shirts sold by Elastic [0]. seems entirely plausible to me that Amazon & Elastic lawyers could get into a pissing contest about potential trademark or copyright protection of that term and whether sending it as an HTTP header constitutes infringement.
I'm sure the lawyers would be happy to spent lots of time hashing out the details, but Sega v Accolade [1] would seem to be relevant and allow Amazon to use trademarks for interoperability.
Though I suppose they're now implicitly making it part of the API, which I think was ruled to fall under fair-use, though I'm not too sure about the details.
No need for the headers just return a 401 or 403 on a GET to '/'.
> If call to '/' fails with 401 or 403 pass the check and show a warning (message will be linked later). This happens if the monitor permission missing for user. The subsequent checks must be ignored.