This definitely will not make a difference. The Slingbox will stream whatever is coming through its video input port. Whether this is a DVR, a cable STB, DVD player, camcorder, etc doesn't matter at all.

If you have a fast Internet connection at home (10Mbps upstream is more than enough) then any performance issues are almost definitely related to the client bandwidth. Many connections are intentionally configured to limit per-session bandwidth to prevent the successful use of streaming connections like this. This is definitely the case on in-flight broadband services, and many lounge connections are just massively over-contended (or poorly designed) which makes anything more than basic browsing a challenge.

The type of connection you report success with - hotel wired connections - is exactly where I would expect it to work well, along with fast office / home Internet connections in other locations. Your experience sounds completely normal.