Assuming that people don't clear their cookies is wrong nowadays. Even when your website is a little snowflake, the rest of the web isn't.
I honestly would recommend a Web Extension that adds your search as a search provider, or something like an Open Search Description.
Also: the landing page of GitHub annoys the shit out of me. Every damn time. I don't wanna register with three clicks if I already have an account.
Make a landing page easier for regular users to login, and prioritize login over signups...otherwise those regular users won't be happy. Any company should focus on keeping the regular users happy (UX wise) and not focus on one time trial registrations that will be a burden on infrastructure.