no. the user-agent might, but there is no transient agreement

They refer to the registration-gated "Developer API"

I'm not sure.

IANAL, but reading the developer policy, reference is made to include the Youtube Developer Site & Services, but is not exclusive of other Youtube API Services.

Whether Invidious uses a Public Developer API, a Broker offering their own API, or a workaround with an internal API seems inconsequential.

https://developers.google.com/youtube/terms/developer-polici...

    #  Client: `"API Client" means a website or software application (including a mobile application) developed by you that accesses or uses the YouTube API Services.`

    #  Service: "YouTube API Services" means (i) the YouTube API services (e.g., YouTube Data API service and YouTube Reporting API service) made available by YouTube including those YouTube API services made available on the YouTube Developer Site (as defined below), (ii) documentation, information, materials, sample code and software (including any human-readable programming instructions) relating to YouTube API services that are made available on https://developers.google.com/youtube or by YouTube, (iii) data, content (including audiovisual content) and information provided to API Clients (as defined above) through the YouTube API services (the "API Data"), and (iv) the credentials assigned to you and your API Client(s) by YouTube or Google."

I'm not sure how I feel about software that sits on this grey line of legality sitting on the front page of HN.

There's still the fact that a contract requires affirmative acceptance. The Invidious devs, as we have seen, have not accepted it.