If the tile is attractive, I would not cover it with anything except area rugs. If you hate it, that's a different matter.

It is not attractive, the tile looks dirty and dingy no matter how much I scrub and the grout is a horrible reddish-brown color.

My tile was looking horrid, always dirty looking. I had someone come in clean and scrub the tile, and apply a different grout color with sealer. Looks 100 percent better. Went with a darker grout, the guy said NO, but I love it, because the grout is dark...hence none of the grout looks dirty.!!!

I would try area rug first over the tiles to cover as much as possible.

More or less, yes. We had filthy hardwood last finished in 1939 and covered since then in asbestos lined linoleum. We took out the linoleum ourselves. The flooring guys put a thin plywood subfloor in and new vinyl (without asbestos) on top of that. Looks great, "walks" great, and no jolt gong from new vinyl back to wood. I'm sure the same could be done with any flooring you choose. Go for it!

We had a mostly tiled house. As long as the tile is level, you can have a floating wood floor (like Pergo) installed over it. We did this and never had a problem.