The best thing to do is not to have it drain under the patio -- when you change the moisure content of soil, it can compact differently, or wash out, both of which could cause the problems you're seeing. If you can, change the drain there so you're using non-perforated pipe, and run it out further to a drainage field past the bricked area.
The problem is, you want to compact the soil for the patio, which will reduce the air voids in the soil, making it less permeable and not drain as well, making the french drain less effective.
As it's bricked, if you slope it away from the house, you'll reduce the amount of water absorbed in that area, which should hopefully eliminate the need for a french drain to collect the water to take down to a drainage area.
If you want to get the best compaction of your soil, dig some if it out, then compact the soil, then add a few inches back in, compact again, then repeat until you're out of soil. Then repeat for the base pack, check it for level and then start laying your pavers.
I assume you are talking about this.
I have seen this product used twice before in this type of application.I have never used it in a french drain, however, I packed some into the end of a downspout so the water wouldn't drill a hole into the ground.
I have also seen it used as a substitute for rock in a drainage ditch along the outside wall of a house. Months later they wound that the rubber was compacting under its own weight and the weight of the dirt above it and began to lose its functionality and actually prevented water from leaving the area. The surface area is too great along lateral edges,(it lays flat) so it begins to clog up.
I assume that your question steams from worrying about the difference between rock and rubber. In your case I would NOT use the rubber because the rubber mulch could settle in the drain and block up the pipe, especially if the mulch is longer than it is wider. I'd use rock. It won't bend or flex, plus it's proven.
Best Answer
Admittedly guesswork, but...
I would expect that, over time, the flagstones would settle into the gravel and mostly stabilize themselves (outside of frost heave and similar issues). There still could be tripping hazards, though, just as there can be with bricks after frost heave and tree roots lift them...
Note that the french drain doesn't have to be exposed through the full 360-degree circuit of the house. If there's a short section where you have a walkway, that probably won't make much practical difference, as long as the whole thing is able to drain properly.