If you saw the power light flash when you reset the SMC, that indicates the reset is done and successful and you can move on to the rest of the steps in the guide.
I would let it charge up for 1 hour and then run through the steps above before getting on with Apple Support to see what the cost for battery service with mail in so you can then decide to go with Apple or look locally for an authorized or non-authorized repair. Apple will be able to refer you to their consultants that are certified / local (or as remote as you are if that’s the case).
I’ve had them overnight a box, get it in the mail, send out the machine and get it back a day later. I’m in a big city, though but mail-in is almost always faster than the Genius Bar for me these days. Even for battery service. Getting the machine to the parts is easier and cheaper than making an appointment and the stores often just send these out so you don’t save time for this repair.
Good luck - be sure to make an answer with what you do - the above is general advice and might or might not be the best for you.
There is almost always enough residual power a day or two after the machine shuts off for a SMC reset and if the battery goes completely flat, that too resets the SMC in a way since it cold starts when the trickle charge returns. Also, be sure you can’t try a different charger if it’s just that.