Search fires on partial queries via `phx-change` before user finishes typing

[NikolaRHristov]

Description

The search input has phx-debounce="300" combined with phx-change="search_changed", which fires run_search/2 on every 300ms pause while typing. A 300ms pause mid-sentence is very natural, causing searches to fire on partial queries like "how w" before the user has finished typing.

Steps to Reproduce

1. Open the search page
2. Start typing a multi-word query, pausing briefly (>300ms) mid-sentence
3. Observe that a search fires for the partial query before the full query is entered

Expected Behaviour

Increase phx-debounce from "300" to "600" on the search input so the user has enough time to finish typing before a search is triggered.

Current Behaviour

Search fires for partial queries on a 300ms pause, resulting in premature results before the user finishes their query.