It's not a silly question at all.
Maybe Sling is just a "view on my computer" solution.
That's pretty much the whole deal, yes. The Slingbox unit is a television without a screen. It uses your computer as it's screen, it's not really designed to send a stream directly to a television.
If you can find one, the SlingCatcher was designed to do what you're looking for and more. That's an expensive way to go though, and it appears that the SlingCatcher is a discontinued item.
If your computer has TV OUT, you could hook it up to a television and do it that way. That doesn't get around the problem of needing a computer in each room, so I don't think that's a useful solution. It's possible though.
Slingbox may not be what you're looking for. Sounds like you might be looking for a TV Extender, something like this or this. Extenders are designed to use a wireless link (usually at 2.4 GHz, 5.8 GHz is better) to send a video signal over the air to a TV without running wires. They all send picture and sound from an A/V source, the good ones also have an IR Repeater built in so you can use a remote control on both sides. They DO NOT normally have tuners built in, so you can't just screw the cable coax onto the back or use an antenna. You would need a source with composite + audio out (the normal RCA red-white-yellow connectors). Depending on distance and the radio environment, the range on these things can be a disappointment.
I used an extender similar to what I linked to in a classroom. We set it up so the TV cart could move around the room but the VCR/DVD combo and the school's CCTV feed could stay securely in my office (part of the same room). As long as I didn't stray too far, it was cool- and no more ridiculous cable runs to trip the cart over with.
What was more cool was taking my Sling AV to school. I patched it into the video sources in my office and got it working with the school's LAN. I had to connect my laptop to the TV cart, but now I could roll the cart anywhere in the building and sling in over the school's wifi network. It was a short-lived experiment, but man did it work ! You want street cred among school staff ? Find a way to pipe March Madness basketball into a teacher's lounge that has no cable.
Hope this helps give you some ideas on how to sort out your problem.