IMO, pipe is cheap relative to the cost of labor and other materials for this project. In our clay soil, it was important not to excavate too deeply and disturb the soil. It was better to dig only deep enough to lay down the filter fabric, put down some gravel in which to embed the drain pipe, add more gravel to cover, and then wrap the whole gravel/pipe with more filter fabric to keep sediment out of it. The best description was that we were making a burrito.
Also, very important to make sure that the pipe is laid with the proper pitch, avoid sharp bends, and be sure that there is an adequate outlet for the water discharge. There is actually some precision required for a good installation if you want it to work well.
FWIW, the so called filter fabric I see at home centers is nothing near the strength of the filter fabric used for road bed construction. That stuff is so tough it's hard to cut. I bought some from a local road building contractor the last time I needed some.