The MAC address will never be seen by an external web server, only the ISP's equipment to which your router connects will know the MAC address of your router. Systems beyond that piece of equipment will only know the external IP address at your end. The MAC address the web server will see will be the MAC address of whatever piece of network equipment sits in front of that web server, since the MAC address is at the Ethernet level, i.e, it appears in an Ethernet frame encapsulating an IP packet. Your IP packets will be re-encapsulated in different Ethernet frames as they traverse the Internet.
If you reboot your router, it may receive a new IP address from your ISP. The IP address on most home users' routers is usually assigned by DHCP, i.e, it is dynamic and subject to change over time as the lease time for the IP address expires or the router is rebooted and requests a new IP address. In some cases, though, an ISP's DHCP server may reassign the same IP address if you reboot, if the MAC address of your router hasn't changed. But, for those ISPs, if you change the external MAC address of your router, then the ISP's DHCP server will assign a different IP address.
It is possible that an IP address that your router received was assigned to someone else previously who engaged in malicious activity that resulted in that IP address being placed on a blacklist, or there could even be a system on your own LAN that was used for malicious activity unbeknownst to you, e.g. if it became part of a botnet. When you changed the MAC address of the router, it received a new IP address not on the blacklist, so access to those websites was no longer blocked, because they now saw a different IP address.
You can determine the external IP address that other systems will see by going to a site such as whatismyip.com that will display the IP address external systems, such as webservers see for network connections from your location. You could try changing the MAC address back to what it was previously, determine the external IP address it then gets, and, if you again have problems accessing particular websites you were unable to access previously, do a search on that IP address with Google or some other search engine to see if you find any results revealing it is on a blacklist. You could also search for the IP address on sites that monitor domain name and IP address reputations, e.g. BarracudaCentral IP/Domain Lookups, though its absence on any particular reputation list doesn't mean it isn't present on some other blacklist.