How to ensure that the bottom of blind baked pie crusts brown properly?
I then chill or freeze the dough 


That's your problem, right there. Especially if you are using a ceramic pan, like a traditional quiche pan. 

For a long time, I also followed the ubiquitous advice to keep pie crusts as cold as possible, even freezing, and struggled with the same problems as you. Then, with more experience and some experimenting, I realized that the "cold is good" rule gets overinterpreted. 

Instead of all this craze of "more cold, more cold", it is much better to use "just as much cold as needed". And what do you need the cold for? First, if you use a "cut the butter into the dough" method for a flaky crust, it is easy for an inexperienced baker to overmix. Cold butter mixes slower, making the process more forgiving towards bad technique or missing experience (to recognize the right moment to stop). Second, if you want to roll out the dough into the pan as opposed to using your fingers to spread it, there is a right temperature range to do that. If your dough is too warm, it won't work properly. Actually, it won't work if it is too cold either, but you seldom hear about that. 

And that's it. No more cooling needed. In fact, I once purposely made crust without using any cooling technique - started with room temperature butter, and the dough and all utensils were kept at room temperature throughout. I placed it by hand into tartalette tins because it was too soft for rolling. And I ended up with perfectly nice flaky crust. So I would say that, while erring on the side of "a bit colder" can be helpful in making the process more fault-tolerant, excessive chilling is not needed. 

Now my crusts are in the 15-20 C range when they go into the oven. And the pan itself is room temperature, it has never spent time in the fridge or freezer. And the problem of "wet bottom, overdone sides" has been greatly reduced. I don't bother to do anything more about it if I'm going to eat the result alone at home. 

If you are baking a perfectionistic pie, after starting at the right temperature, you have to stage your blindbaking. First, you bake for some time with the beans, until the "walls" are just firm enough to not fold in the heat. Then you remove the beans and the foil. If you want to fiddle, or left it a bit too long in the oven, add a strip of foil over the walls. Now continue blindbaking without beans, until the bottom is well baked. The walls will still be somewhat more baked than the bottom, but the difference will be tiny. Now you can continue with filling, etc.