My parent's home, circa 1930s have the clay pipes running underground and were hooked up to the various downspouts to shoot the water well past the foundation and out into the street. They've been functional since the house was built, but probably remained so because we snake them out with a drain snake occasionally. Yours could have been a French drain, but if they were they'd not have been solid clay, but have some perforations for water to percolate into them to be carried away. If you have had water issues with your basement, part of the reason may have been non-functioning drainage tiles. If they are hooked so water flows into them, if they are clogged, or particularly if they are fractured, you'd get seepage next to the foundation. If they are only buried six inches deep, then if you find areas of wet pools or foundation leakage it shouldn't be too much of a problem exposing that particular area to check for patency. Ten inch sounds very large for that purpose. The ones at my parent's house are perhaps six inches. They were once used to funnel runoff to a huge cistern. Have you found a cistern on your property? You may have one hidden somewhere.

I'd be surprised to find a cistern on the property. The lot slopes down at a pretty steady pace. Also, the clay drains seem to end right past the house.

Yes, the drain is clogged. The prior house owners didn't do much upkeep, so I suspect there is quite a bit of sediment in them. However, they are likely a secondary culprit at the moment. It looks like an old retaining wall on the property line altered the grade. The leakiest section was actually sloping towards the house. After a weekend of shoveling/hauling dirt, I think I have that part fixed.