I imagine the two ways to read the Rema are as follows:
1) Instead of being mekayem the chiyuv of ad delo yoda by getting drunk, you can be mekayem the chiyuv by sleeping, in the state of which you are lo yoda. This does not patter you from the chiyuv of drinking, which you are mekayem by drinking yoser milimudo.
2) Instead of being mekayem the chiyuv of ad delo yoda by getting drunk, you can be mekayem the chiyuv by falling asleep through drinking. If you consume enough wine to cause you to sleep, you are mekayem the chiyuv.
In the second reading, you cannot just go to sleep; the sleep must come through the drinking. This is possibly what Halevi meant. Although I can see why you could read this into the Rema, I don’t understand why this should be the case.