Search the Internet a bit. The white pine could be diseased or the yellow needles could be caused by pests. According to Forest Service.edu: "Pine needle miner larvae feed inside needles
causing them to turn yellow and dry up"

Search for yellowing "weeping white pine" and google Images also for "common diseases for weeping white pine" to ID and narrow down the most likely disease or horticultural problems. Good luck!

If there are no pests or microscopic infestation visible that can be identified, then other common causes of stunted growth (and yellowing) often are associated with water-logged or alkaline soils (or soil pH outside the weeping white pine's preferred range).

Depending on care and the sequence of events and observable changes, poor root growth, stunted growth, and yellowing are also common symptoms of herbicide and insecticide damage (and accumulated).

Watch out for high sodium and chloride salts from winter road salt spray and the organic amendments you are using. Pinus strobus is sensitive to salts and likes rich well-drained acidic soils (just like gardenias).

If plant growth is stunted but without identifiable pests or plant disease, the likely problem could be an alkaline soil high pH problem, especially if it was grown on heavy poorly drained potting mix and now with added organic amendments which can all cause chlorotic and nitrogen dificiency as well as contribute to soil alkalinity (and possibly your municipal water as well) that will interfere with plant's uptake of iron and other minerals. I successfully used agricultural sulfur to lower the pH of my potting mix for my gardenia because my gardenia was not iron deficient primarily but deficient in nitrogen and other minerals; however, I also used a light liquid iron foliage spray to keep the secondary macros and other micros balanced somewhat when correcting nutrient deficiencies and stunted growth problems in a speedily but safe approach. Generally, a few repeat applications over months of a minimum of 6% iron chelate is suggested to reverse a chlorotic and nitrogen deficiency, but consult resolutions suggested specific for your white pine cultivar after you've identified the likely problem(s) relevant to your white pine first. Use Sphagnum peat moss and composted pine bark fines will keep the pH on the acidic side. Research to see if you need to use special fertilizers (for acid-loving plants) or treat your water to maintain potting soil pH. Here is another credible link about caring for Pinus Strobus 'Pendula'
at, Ohio State University

Thanks, Legacy, you have given me a lot to think about and much to do. I appreciate your advice.

I examined the pine tree and could find no sign of obvious pests or diseases, no webs or black dots or spots on the needles, etc. The ph is slightly alkiline, and is higher than the preferred range I found for pinus strobus, which is supposed to be slightly acidic. I'm going to try lowering the ph and see what that does.

Thanks again for the suggestions, Legacy.

Bad news for me and my maple. It suddenly began to look worse over a period of two weeks or so, so I decided it was time to throw in the trowel, so to speak. While disposing of the tree I noticed that there were just none of those fine roots or root hairs, and in spite of the fact that I had purposely changed the soil to a good-draining medium, the container was a big soggy mess. We've been having one of our occasional wet summers, lots of frequent showers. I'm thinking root rot just killed the poor thing off.