It sounds like your esc is fine. The receiver seems to be dead, and that is not uncommon. Receivers die sometimes. Fortunately, you can get another receiver for around $9 USD. Look on Amazon. That would be my first step, because the GR3E receiver will light up the LED and it will flash when there is no radio signal to it, regardless if it is bound or if your esc is calibrated.
That button on the esc is to program it. Not knowing which esc that is, you'd have to do some looking around to find the manual for it to know how to program it. But a lot of them are typically the same. If you look at esc's on banggood.com, you'll likely find that esc. Then you can hunt for the manual for it on google. So since it should already be programmed, I'd avoid pushing that button til you know the procedure, as you could inadvertently reset everything, then need to find the manual to reset it.
Edit...
I think that ESC is the Reely H0050. I did a quick search for the manual and found this...