Here are my thoughts -
1)Just because it is your third stay at this hotel does not mean they recognize you, or even if they do, remember that you are Platinum.
2) Arrival gift points often do not post. OFTEN. The fact that they did not is not "proof" that your Platinum number was or was not attached to your reservation.
3) You are definitely NOT entitled to the missing last 2 arrival gifts. The rules clearly state that your claim must be made before you check out. No exceptions. Call CS, get your points added, and learn from this mistake.
4) Re calling the front desk and making your reservation - read point 1. ALWAYS ALWAYS ALWAYS ask if they have your number attached to the reservation.
Quite honestly, I am guessing that your Platinum number wasn't attached to the reservation, especially if you made it verbally and "assumed" it was added.
Finally, and this is strictly personal, there is no way I would agree to be walked to the Extended Stay America. Anywhere. If I had to pay for my own room and then take it up with Marriott later I would do so.