Even tho some privacy respecting search engines like brave and startpage too showing me captchas.

I’ve never had a captcha with DuckDuckGo, if you want to give that a try. Otherwise, metasearch engines like SearXNG act as a proxy between you and other search engines.

From my search i finded that adguard or any other dns servers establish dnsotls-ds.metric.gstatic.com this connection in order to check the status of the private dns enabled or not. To block this i have to use a no-google blocklist which leads to inconvinience.

Good to know. It’s up to you whether you want to trade privacy for convenience.

No gecko based android browsers provide option to change dns provider.

GrapheneOS’s browser Vanadium is a good option if you want to move away from Firefox-based browsers, but it’s not easy to install anywhere other than GrapheneOS. If you’re up to try, here’s how.

Brave is making too much background connections which is annoying.

Brave can be hardened to minimize most of those, but I agree it is annoying that there are still background connections.

Also it would be nice to know leaking my location to dnsotls-ds.metric.gstatic.com leads to any consequences. Or is it just a private dns current status checking url ?

Besides Google being able to see every time you ping the domain, there’s not much else going on. It’s unlikely that it’s leaking any private data, so it’s relatively harmless. It’s not ideal that it connects to it, but it doesn’t pose too large of a threat.