I would like to check the availability of Internet connectivity by sending ICMP "echo request" messages to some IPv4 addresses. Those IPv4 addresses should be as reachable as possible in stability wise and they should not handle ICMP traffic as low priority. I know that DNS servers of tier-1 ISP's(e.g. Level3) and large content providers(for example Google 184.108.40.206) are a popular choice. Maybe it's wiser to use some service which is using DNS entries(IP addresses in different ISP networks) in round-robin fashion like pool.ntp.org? In a nutshell, which IP addresses do you use for this and why?
An upstream router at your ISP might be a good choice too. Typically something one hop up from your local router.
Though a word of caution, the problem with doing this or using Google or L3 would be that it is possible the responders not to work, but the connection still be up and online. Just make sure you use them as a test and not rely on only one as an automated way to failover a connection... One of those, been there, done that situations...
As you mention I primarily use googles public DNS. I have never had an issue with it, however if you feel like that doesn't cover it, you could always check multiple hosts. The chance of 2 major datacenters being down is very little.
For example :
b.resolvers.level3.net - 220.127.116.11 google-public-dns-a.google.com - 18.104.22.168