It's a shame that we don't have any standards for the concerns of canonically testing Internet reachability and for authoritatively redirecting network users to a captive payment portal (without having to resort to ugly hacks that often break with TLS).
But even just having an RFC to yell about is great, thank you :)
So yeah I would pick something simpler for a network access test probably?
It's a static page that every Apple device relies on saying only:
<HTML><HEAD><TITLE>Success</TITLE></HEAD><BODY>Success</BODY></HTML> ~$ ping -c1 1.1
PING 1.1 (1.0.0.1) 56(84) bytes of data.
64 bytes from 1.0.0.1: icmp_seq=1 ttl=56 time=7.89 ms
--- 1.1 ping statistics ---
1 packets transmitted, 1 received, 0% packet loss, time 0ms
rtt min/avg/max/mdev = 7.888/7.888/7.888/0.000 ms