Landscape fabric, which is typically a porous, woven material that allows water penetration.

As you are discovering, roofing felt/tar paper isn’t water permeable. Also, depending what your roofing material is made from (typically asphalt), I’m not sure I would be comfortable with the potentially toxic/harmful runoff in a children’s play area. In addition to traditional landscaping fabric (which tends to not hold for for very long), there is also permeable landscaping fabric that would be ideal (longer “life”) - not sure if this is something you can get/order from Home Depot/Lowe’s etc.

I'm going through this process right now and suprised how expensive the site preparation is! I'm following this guide as much as possible: https://www.cpsc.gov/s3fs-public/pg1.pdf 

Instead of ripping up what you have already laid down why don't you punch some drain holes in the roofing material.

But if you use landscape fabric, get polypropylene FABRIC ... not the cheap stuff that tears apart. Also, you must excavate and bury it, pin it down and then replace the mulch. I would not use the ground tires. They stink when hot and wet. So much simpler when I grew up. We had lawn and then we wore a hole in it and had dirt. We grew up and our parents got their lawn back. It's nice to not need to mow around all the equipment. But then you have to do the excavation, fabric, the mulch and some edging to hold it in. It would be a rare parent who would do all that in the MCM days, if anyone had ever thought of it. Which they didn't.

If you are using the rubber mulch for safety reasons such as falls from the play structure, it should be 3-6 inches deep. Less will not provide a fall "cushion" and you might as well leave the grass and dirt underneath the play set. Consider recessing the mulched area below grass level or edge the area with something, plastic edging, wood, etc If not, the rubber will get kicked into the surrounding grass and it will be a lawn mowing nightmare. It's a great looking play set and I'm sure your children will enjoy it.

Years ago, many years ago, we put ours in a giant 16X24 ft. sandbox. The swing set wore out and was removed. The boys grew up, they and their friends played in it at their high school grad party. Great fun. Oh, the pics!

I didn't think that recycled tire crumbs were safe for playground use. All kinds of nasty chemicals in tires.

Don't worry about mold and bugs The puddles will dry long before any mold grows there - it takes awhile for mold to set in. Bugs won't live there either - there's nothing to eat.

Great play set.

I would worry more about the rubber mulch than mold. As it is currently set up, the chunks will end up in the lawn because they will be kicked out by the kids playing, and I too would worry about chemicals and steel belting around the kids as well as ultimate disposal of the material when the playset is no longer needed.

Wonderful playset, however.

Artificial turf fields often used recycled tires, sprinkled on top. Many youths have been diagnosed with respiratory problems, thought to be caused by breathing in the recycled tire dust.

Agree with landscape fabric, recommend the good stuff, not the cheap. Recommend replacing the recycled tires with wood chips.

Soccer organizations are doing research on the recycled tire dust. Two of my nieces played division 1 soccer, one of them is currently a part time high school coach. One of the players my niece coached was featured on the national news, she has a respiratory problem, thought to be caused by the tire dust. She is playing division 1 soccer. Goalies have more problems because they spend more time on the ground than other players.