First, WSUS NEVER pushes updates. WSUS is a pull system. You approve an update, it sits in the repository until the client agent pulls the update to itself. You "use WSUS to distribute updates" is a more correct way of saying it.
I would recommend that you CONTINUE to use it.
It does not cause problems and always starts with the WSUS server download initiation first, then it moves to 'hey, does anyone here have this file I can download from you?'.
The issue with your internal website becoming very slow would have a different reason. Is it housed on the same server as WSUS? If yes, then you're saturating the Ethernet line with traffic from both websites - your internal website and the WSUS Administration website. Allowing DO to properly fetch updates from peers would actually be a benefit in this case.