I also have had logs for a long time. In regard to Fungi Perfecti versus the rest of the world, I find that Fungi has more productive shiitake strains, whereas others have less productive but better tasting strains.
Shiitake is king with plugs, I agree. Oyster is more difficult to flush at will, and other species are more difficult to grow. You have to cut in winter, which the OP did. But to inoculate you need high moisture and decent temperatures. I don't know which zone you are, Ampersand, but in Zone 6 I would pile them in the garage, inoculate them in early April, while keeping them moist and under plastic. After a few months the mycelium has taken and you can put them outside, but you could also keep them in the garage until Fall, so long as you keep them watered. Oak is best with shiitake.
Most of my logs were cut in November, injected two weeks later, and placed in 55 gallons bags in my basement, with a hole for occasional watering. The bags limited the need for watering. Some mold would form but not threatening for the fungi. Then placed outside in April-May the next spring.
We try to eat as much as possible when they flush, then dry the rest. We dry and grind all stems, which are on the fibrous side, but dry very well if you just leave them on a counter long enough. The resulting mushroom powder stores indefinitely and is routinely added to soups.
I have a question about what you do to flush them. I used to dunk them into a 55 gallon drum for 24 hours, held down with bricks. Eventually it developed a leak, so now I set a sprinkler in front of them and let it go for 24 hrs. Not as good, but also a lot easier on my back as I do not have to move them.