Is it possible to be a productive programmer with slow typing speed? Yes. I have met some.
But…can fast typing speed be an advantage for most people? Yes!
Like you said, once you come up with an idea it can be a huge advantage to be able to type out that idea quickly to try it out before your mind wanders.
But also, I use typing for so many others things: writing Slack messages and emails. Writing responses to bug tickets. Writing new tickets. Documentation. Search queries.
The faster I type, the faster I can do those things. Also, the more I’m incentivized to do it. It’s no big deal to file a big report for something I discovered along the way because I can type it up in 30 seconds. Someone else who’s slow at typing might not bother because it’d take too long.
Can you elaborate on what happened when you tried to search? I’ve never had trouble.