Small blobs of tar on a (5 year old) flat tar and gravel roof

EPDM rubber roofing might work for you. It can be installed without any heat or fancy tools; I'm planning to use it for my flat-roofed concreted detached garage. It should withstand ponding water, and because your project is small and your playhouse probably doesn't have plumbing, you should be able to avoid seams, complex flashing, and pipe penetrations. You can install the rubber using an adhesive directly over clean plywood.

It can accommodate a floating deck. I have the book Roofing, Flashing & Waterproofing (For Pros By Pros), which includes a nicely illustrated article on installing EPDM with a floating deck. The method they recommend is to use a non-adhered strip of EPDM underneath the decking sleepers, so that any motion in the decking doesn't stress the roof membrane much. You'll still want some slope in the roof for better drainage, but you can compensate for that in the sleepers, allowing you to have a level walking deck.

Lead makes a good roof, but finding leaks in one (or any flat roof) can be a real bear as the leak below may be a long way from the leak above.

If the folded seams are (as I suspect from your description) standing up, except where walked on and flattened, then yes, the walked-on parts would be prime suspects for leakage. Sometimes the the seams are all folded down and soldered, but standing seams are not usually also soldered (it's a fair bit of fiddly torch work.)

Look very carefully for cracks right next to the flattened seams. If you don't see any cracks, run a torch over the flattened area (trying to anneal the metal before reworking it) and carefully stand the seam back up - look for cracks on the lower side as well. If there are cracks you (or a lead roof specialist you hire) can clean the area and solder them.

If there is a wall above the flat roof area, also look very carefully at the flashing where the wall joins the roof - likewise look very carefully around any penetrations in the roof.

Some folks will no doubt tell you to reroof, since lead is "old-fashioned" and therefore also "old" - depending on the shape the roof is in this might be the right thing to do, but both lead and terne-plate (lead coated steel) roofs can last for hundreds of years, so it is well worth trying to fix it first, IMHO.