O2 absorbers do their job and they're done. Life is over. So they will reduce the oxygen in the bucket, but won't do anything about the inside of the packets. Those packets are lined with a plasticized coating, slowing but not completely preventing oxygen from leaching in or leaching out once the oxygen level in the bucket is lower than the oxygen level in the packets.
But the leaching out of oxygen will not occur within the time frame the O2 absorber is active. That will be a slow leaching, it will eventually somewhat lower the oxygen level inside the packets, and outside the packets, the oxygen level won't affect food quality so that oxygen will be somewhat harmless to the food. You'll never get an O2 level as low as you would without the packaging being involved but you will likely get a better result than you would without the mylar and O2 absorbers.
NoFly finding the packets 'rock hard' is telling, but I'm not sure which thing it's telling. Either there is some residual moisture due to the seasonings in the packets or since the packets don't give a very good barrier and humidity leached into the packets. I actually think it is more likely residual moisture in the packets than humidity leaching in because humidity inside the house is more controlled when using air conditioning and heating a house.
The oatmeal companies are not concerned about how much longer their product lasts past the expiration date, they have to test to get those expiration dates and once it goes a year, they aren't going to invest any more money or time into testing or packaging improvement. Long term storage is not in their best interest. I would love to know the percentage of food products in this country that are thrown out and replaced because of that magic date.
My guess is with proper packaging you could get about 5 - 7 years out of the packets stored like you are thinking, more on the five year end of the scale. BUT (and that's a big but) that is a Wild Ass Guess on my part, I won't even pretend it is scientific.
The part of the oats lasting that long is really not a question, it's the seasonings. The only part of the oat subject to going rancid is the germ, where the oil is. The instant oatmeal packets are made with the tiny pieces of oat that flake off in processing the other oats, therefore, won't have any of the germ in them.
And that's all my brain spit out about oatmeal packets... woke up this morning with that info sitting there and thought I'd share it for it's limited value.