[P1] UI Bug: CLI dropdown in agent app

[wendywilsonPNW]

I noticed sometimes the CLI type dropdown appears in the agent app, I think we should also remove the dropdown carat there?
CC <@U05LT41HN4U> <@U08QB9LBTR9>

Slack Message

[jo-oikawa]

Marking as ready for engineering as this just shouldn't flicker on reload

[jo-oikawa]

Screenshot of the dropdown appearing in the input box. Feels like a clear bug on Agents app, unrelated to UI bugs and polish.