I have a RB2011UiAS running 6.37.3 setup as a basic SOHO router. I’ve got ether1 as the uplink and ether2 part of a bridge named local. Everything is working great at this point as it’s a very generic setup.
If I create a default EoIP interface with no connection and add it to the local bridge, I start having connectivity problems. Trace routes and pings are fine, but websites are timing out (speedtest.net, cnn.com). As soon as I remove the EoIP interface from the local bridge, the same websites load without problem.
At an educated guess, check the MTU as EoIP will add an overhead to the packet (forgot exactly how much from memory but a quick google will tell you).
Personally from what your describing it sounds more like an MTU issue to me especially with the sites that you mention i.e. speedtest.net as its very fussy and also most ssl websites are also.
HTH
p.s. I don’t have any real experence with EoIP but I do with VPN’s and this is the most common cause of problems that ive found over the years and your problem sounds almost identical.