What helps against astringent mouthfeel from spinach or chard?

Blanching briefly does an excellent job of neutralizing bitterness, or more precisely, astringency, even if the blanched spinach is recooked.

Basically, boil water, add washed spinach until submerged (5-30 seconds depending on preference; chard could go up to a minute or so depending on the target texture desired). Drain quickly. Submerge drained spinach in a bowl of ice and water to rapidly chill. Press spinach to remove excess water.

Recooking, even in applications like quiche, will do a very nice job of retaining the bright green color and generally astringency, if present at all, is minimal. If I am not recooking in some way, I often just dress simply in something like fresh ginger and soy sauce or a ground sesame, mirin and salt based sauce without reheating.

A remedy I would recommend would be a bit of butter or oil, just enough to give a bit of a coating. You could do classic with a bit of butter, salt, and pepper. You could do Italian with some olive oil and garlic. Or Asian with some sesame oil and soy.

Spinach and Swiss chard contain small amounts of oxalic acid.

This is the same substance that makes rhubarb so tart--in fact, it is the active ingredient in some cleaning chemicals, like Barkeeper's friend. While toxic in large quantities, you would have to eat a lot of greens (on the order of pounds to kilograms) at once to have any issues other than the unpleasant taste and mouth feel.

While you could chemically neutralize it, as with baking powder, the results would probably be hideous on many levels, including strange color changes.

I would buy only young spinach leaves for short, fast cooking methods; or use one of the longer braise type preparation methods.

Though the adding-fat answer is the correct one in my mind (have had plenty of puckery young-spinach sauteed), funnily enough vinegar or lemon replace one pucker feel for another. Growing up, vinegar was always on the table with greens.

I particularly like Chinese black vinegar (from Shaanxi) for tougher greens and balsamic for tender ones

I learned a little trick for this while living in Italy. When you are cooking the spinach, add a bit of milk or heavy cream, just enough to coat it lightly and cook off. I use about 2 tbsp for about 6 oz of raw spinach. Alternatively, I have soaked spinach in milky water before cooking. Rather than patting it dray or straining it, I use tongs to pull the spinach out of its bath, which leaves just the right amount of the milky water on it. This solves the aftertaste problem every time for cooked spinach. I'm at a loss for raw!