What temperature should a crock pot slow cooker cook at

It sounds like you were expecting slow cooking to be like sous vide. Well, it's not. The point isn't controlled sub-boiling temperatures, it's something on the border between simmering and boiling for foods that just need a long time to cook at that approximate temperature.

Slow cooker recipes are not supposed to be very sensitive. They're expecting to be approximately boiling for most of the time, and the difference between low and high is pretty much whether the boil is marginal or a bit more substantial. In many cases, this just matters because the quantity in the pot varies, and it takes more to keep the stuff at the top hotter if it's farther from the bottom, or if it hasn't all cooked down into the liquid yet. In either case, it shouldn't be a full rolling boil; it's just boiling on the bottom, so the rest of the liquid is probably a bit below the boiling point. And yes, this is still slow cooking. It's not boiling fast enough to lose a huge amount of liquid (or worse, boil over) with the lid on.

I wouldn't really try to assign temperatures to slow cooker recipes. Like I said, they shouldn't be that sensitive. If your bean soup recipe didn't work, maybe it was a bad one. But "reliable" in the context of slow cooker recipes doesn't mean "exact times and temperatures". They're generally things that will be perfectly fine if you cook them 25% longer. Not everything in the kitchen has to be precise and formulaic; slow cookers and slow cooker recipes take advantage of that fact. And even if you do try to calibrate, you'll have trouble, since there's a temperature gradient from bottom to top. Unlike sous vide, a slow cooker is not constantly well-mixed. In equilibrium it'll be boiling at the bottom, and 10-20 degrees cooler at the top.

If you really wanted to use a sous vide controller, I imagine something like 95-98C would work for basically every slow cooker recipe, no matter whether they say high or low. Of course, mixing thoroughly enough to make your controller actually work, you may be overstirring whatever you're cooking. But the point is, things you cook in a slow cooker aren't really going to care much what the exact temperature is; it just matters that it's hot, near boiling, and not boiling so fast that it sticks on the bottom or loses a lot of liquid.

Finding reliable recipes... Well, it's like anything else. If you're looking on the internet, you have to learn to judge for yourself and look for warning signs, or stick to sites with lots of reviews. You also have to accept that sometimes you have to test for doneness and be flexible about time. This isn't really unusual; baking recipes should always have some kind of test ("until golden brown") and the actual baking times will vary. (With something like bean soup, sure, maybe the recipe was bad, maybe you didn't soak enough, maybe the beans were a little different. A stovetop recipe wouldn't have been precise either.) If all that isn't good enough for you, buy a slow cooker cookbook; tons of those have been published in recent years.

Without any other information, it sounds like you have to assume that your slow cooker has only a warm setting (labeled "low") and a high setting, and use high for cooking. (Usually they have warm, low, and high.)

But it's possible that they're just being overly paranoid in the manual. I would personally try heating something on low (possibly just water, as a test), and seeing whether it boils or at least reaches and maintains a decent temperature.

The thing you're trying to avoid is the food spending too long in the "danger zone", below 140F (60C). If the low setting only just brings it above that, then it's going to take forever to heat up past it, and your food will be unsafe because all that time it wasn't hot enough. But on most slow cookers, the low setting brings it up to at least around 180F and on many, it even very slowly boils. If your heats that well, you can certainly use it for cooking. The one thing to keep in mind is that if it still takes a long time to come up to temperature, you may want to start it on high to get it all heated, and then switch to low for the rest of the cooking time.