What is your definition of "do not work" ? Do you get nothing at all ? Part loading ? Problems with transactions ?
Is it that they use an unsuitable encryption protocol
Nope.
does the problem lie in their baseline computers ?
Each site will likely have a different architecture, different server software, different load balancing mechanisms, different security.
It is a possibility that if you are having issues with transactional aspects whilst connected on a slow connection, that you may be hitting timeouts configured somewhere on their infrastructure.
The fact is that with the increasing ubiquity of broadband connections, the chances of developers not remembering the lowly narrowband user is becoming increasingly high.