Ah you're quite right (I was expecting that the queries would be POST-like, not GET-like).
This distinction doesn't matter too much though, because having the browser be responsible for this fails in at least two ways:
1. The browser shouldn't need knowlege of how to parse every query language of every webapp you use.
2. The browser can't actually give you the results of those queries. If I search for "tps reports", the browser knows I entered that query string, but it doesn't know what the 15 results were, and so can't, for example take my prior search into account when I search for "12/17/2022" to uprank the TPS report for that date, as opposed to some other report. Nor can it even do the weaker bit and assume that since I searched TPS yesterday, when I search "reports" today, it should consider upranking tps reports.