I agree with your points in the first sentence here (URL/serch bar) etc. However this is by design. It protects Google search with a moat, which in turn protects the lucrative Apple-Google search deal. Microsoft (with Edge and Bing) complete the three-party monopoly in OS-browsers-search [0].
Yes, we need good alternative browsers and browser innovation. There are plenty around, but none of them apart from Firefox (and forks) support search diversity, as they do with the search box [1].
Our own, search engine app, has a way of supporting similar one click search choice: Search Choices [2]. Users love it. Why be bound into the one-search-to-rule-them-all paradigm?
But watch out, browsers are being circumvented. The moat now being built is around how search links with the operating system and voice assistants. How do these three companies play? And do they play fair? Who cares you may say. As Tim Wu has said [3] "it is not a crime to be a monopolist; it is a crime to abuse your monopoly power." How will he view how they play together?
Microsoft ignores your choice of search engine in Edge with searches from Windows [4]. If you have chosen Ecosia, or set-up Ecosia as your search preference in Edge, that choice will be ignored and Bing will be used. Your choice is overrided.
Apple does something similar with Spotlight [5]. Suppose you chose Mojeek or Ecosia as your search preference in Chrome. And Chrome is your default browser on your Mac or iPhone. If you have Google as your search preference in Safari (and remember that is the default which Google pays $billions for), then the Spotlight directed search will be done with Google search in Chrome, not Mojeek or Ecosia. Your choice is overrided.
Google have Android. What can you do with the Search widget?
[0] https://blog.mojeek.com/2022/05/gatekeepers-of-the-western-w...
[1] https://blog.mojeek.com/2020/12/popping-filter-bubbles-in-fi...
[2] https://blog.mojeek.com/2022/02/search-choices-enable-freedo...
[3] https://newrepublic.com/article/111650/why-does-everyone-thi...
[4] https://blog.mojeek.com/2021/11/how-microsoft-sucks-people-i...
[5] https://twitter.com/ColinHayhurst/status/1533854896633544704