This is clever and cool, thank you for building and sharing it. Any chance it supports JSON for machine detection? Similar to jsonip.com
Unfortunately it's not possible to check for CGNAT via web browser. The only way to is to check the WAN IP directly on your router.
So it doesn't check if you can actually host it (because it can't check, there is no other way than checking on your actual network)
Literally all it does:
if org belonging to ip contains wireless, mobile, cellular, verizon, at&t, t-mobile, sprint: IP is cellular (really bad way to determine if an IP is cellular or not by the way)
if IP belongs to asns 701, 6167, 22394, 21928, 23402: IP is cellular
if IP belongs to asns 7018, 20001: IP is not cellular
if none of above if statements match: IP is not cellular
if IP is cellular: no you can't
if IP isn't cellular: yes you can
That's it, that's the entire site.
And saying "Public-looking IP" doesn't make sense, you'll never get anything other than a public IP from icanhazip (or any other ip fetching service). Also having a cellular IP doesn't automatically mean you are unable to receive inbound traffic, some providers sell "mobile broadband" that allow you to open ports, etc.