Earning calories with exercise: to eat or not to eat...?

I eat many of mine back. The actual percentage depends on several things: how hungry I am throughout the day, what my workout schedule looks like for the NEXT day, when my next low-calorie day falls (I try to have one low calorie day per week), etc. Most important is when my next big workout is. Today's calories provide fuel for tomorrow's workout, so if tomorrow is a hard day, then I'll eat most of today's workout calories back.

I do not eat them back
I use an HRM for a more accurate reading of calories burned. I never use MFP's database.

I think it depends on how much you are eating in the first place and how many calories you burn in your workouts.
I never used to eat my exercise cals back, but I started plateauing so after advice on here I started eating them back.
Sometimes I don't manage to eat them all back, but I attempt to net 1200 per day, and if i don't then i'll give myself a treat meal at the weekend.
For example, the past two days with exercise i've only been netting around 900, so i'm going to treat myself to a Nando's this weekend

You should be eating at least some of those calories back. They're calories that your body is going to need if you want to keep exercising with sufficient energy and health to do so (which also helps prevent injury). You can skip it to lose "faster" if you'd like but you'll lose some energy and ability to exercise. Do you want to just be skinny or would you like to lose weight and be fit?

MFP is set up that you are supposed to eat them back. Your daily goal already has a deficit built in, meaning eat to goal every day, do zero exercise, and you'll lose weight. Burn off cals through exercise and you're creating a much larger deficit, which is not good for ya, and can cause more harm than any good in the long run (stall in weight loss, no energy for workouts, burn out, screwed up hormones & metabolism, loss of lean muscle, body wants to store fat, etc).

That's why MFP adds those burned cals back into your goal when you log your exercise - you are supposed to eat them back to bring your NET calories up to goal for the day. Food is fuel! Can't expect that furnace to keep burnin' hot if you don't fuel it!

Edited to add: This system works well assuming you have set things up correctly - meaning being honest and realistic with healthy weight loss goals (as in not 2lbs a week if you've got 20lbs to lose), activity levels, etc.

I wear a HRM when exercising. I usually eat back the aerobic calories burned but not resistance training calories since those are not accuaretly recorded using the HRM. I just use the HRM as an over gauge of how hard I worked out doing resistance exercises for comparison. I think you should eat at least some of them back as a "reward" for the hardwork. But reward still needs to be healty and appropriate for your diet plan.

Eat half of the calories you've earned.

This is a good reward for exercising and you will still lose more weight. Double win.

Do people really believe that MFP is so poorly designed that it adds these calories to your daily goal for no reason?

Yes, if you set your profile up accurately and are using MFP as it was designed you are supposed to eat them back, within reason and taking into account any inaccuracies between MFP's database and your actual workout.