That "flimsy wood paneling" is called a "picture rail" the first link I found describing it, not selling it. Take a look at pictures of the inside of a castle/palace somewhere (Windsor or Buckingham, for example), you'll see that all the pictures hang from picture rails because the buildings are made of stone.
Its entire purpose in life is to hang things from it. It is there precisely because you have some sort of brick/plaster/concrete walls that are difficult to fasten things to. Now, you have one thing that was fastened to this hard to work with wall construction, and have complete freedom to hang things as high or low on the walls, anywhere you want.
Learn to love it and make it a conversation piece in your house by getting some picture rail hooks (you'll have to look at the top of your rail to see the profile & size), some picture wire, and learn how to work with it.
The one thing you may need to do is to learn how to install more picture rail (I'd recommend matching the profile and height of the existing rail, just to make it look planned, not haphazard) if you want to hang things on other walls.
Note that if your existing picture rails are as flimsy as you think, you may need to remove them and replace them with something beefier. I'd suspect, though, that they're stronger and more firmly attached than you think.