Indeed! It took me a few seconds to realise that it was 1.1.1.1 that's the problem. When I could connect to my favourite websites from my mobile phone (with default DNS settings) but not from my laptop (with 1.1.1.1 as the DNS), 1.1.1.1 became the obvious suspect.
$ ping -qc4 1.1.1.1
PING 1.1.1.1 (1.1.1.1): 56 data bytes
--- 1.1.1.1 ping statistics ---
4 packets transmitted, 0 packets received, 100.0% packet loss
But the Internet connectivity was fine, in general.
$ ping -qc4 8.8.8.8
PING 8.8.8.8 (8.8.8.8): 56 data bytes
--- 8.8.8.8 ping statistics ---
4 packets transmitted, 4 packets received, 0.0% packet loss
round-trip min/avg/max/stddev = 14.075/16.401/18.641/1.895 ms
The very next thing I did was to come here on HN /newest list to find out if the community is talking about it and here we are!