Android browsers can provide what are called “Custom Tab Intents”, meaning that other apps can request to open web pages in a webview-like custom tab. This depends on browser support, with WebView being the fallback when your default browser does not support it.

If you’re looking for a chromium based browser, I like Cromite (successor to Bromite). Otherwise, Iceraven and Mull are two Firefox forks for Android that work well in my experience.

Graphene only supports Pixel phones. You could try to port over the device tree, HALs, and build system changes from LineageOS, but you would have to maintain it all yourself and run your own builds. Im using a pixel and I really enjoy graphene, but I’m not sure if it’s worth that much effort

I selfhost SearXNG and route it’s traffic through a VPN. I rarely get blocked by search engines, and when I do I just change the VPN server. You lose the benefit of your data being intermixed with everyone elses on a shared instance, but you still get the other privacy improvements from SearXNG