Before the Normans brought in all the laws restricting hunting, would it have been particularly normal for English commoners to supplement their diets by hunting "for the pot" (IE the same way American frontier settlers later would)?

Hunting was not a common source of nutrition in Anglo-Saxon England, no.

Wealthy Romans hunted, but hunting essentially disappeared from Britain after the collapse of Roman estates. Less than 1% of the animal bone found in trash pits from 5th and 6th century sites is from wild animals, and almost all of this <1% is antler (antlers are naturally shed each year, and can be gathered in the forest without hunting the deer from which they came). If deer were being hunted in these centuries, they were not being eaten (and their bones discarded) at home.

This changes in the seventh century. Here we start to see big elite halls developing, and these elite buildings' trash middens contain a small -- but now verifiable -- number of wild animal bones, mostly red deer. The skulls were most common, while the meat-bearing bones were more likely to be found distributed around the big houses, in smaller dwellings. This suggests that the guy in the big house -- the lord (lady) -- was keeping the trophy (the skull) and doling out the meat among his (or her) supporters. The overwhelming number of bones at all these sites remain, however, domestic (beef, sheep, pig, etc).

By the end of the tenth century, this has changed again. Now, all the deer bones are found at the big house, and the ordinary people are just eating domestic meats. Other contemporary evidence suggests that the sharing economy, and the sense of community that went with it, was being replaced by a new aristocracy that was very powerful, and increasingly set apart from the people it ruled. Hunting became -- as it had been for the Romans -- a way this aristocracy marked its difference from ordinary folk. Incidentally, it was at this time -- not after the Normans arrived -- that the first elite deer parks appear to have been established.

When the Normans came, they invested heavily in deer parks, and they introduced new hunting practices, together with a new kind of deer (the fallow deer; though there are some questions that fallow deer may have been introduced earlier, by Saxon elites). In doing so, they were building on -- not overturning -- the norms established by their Saxon predecessors.

Why did ordinary people not hunt deer? By the Norman Conquest, deer hunting was an activity guarded by the elite, but earlier -- say, in the sixth century -- there weren't social controls to prevent a peasant farmer from hunting deer. So why didn't they? It may be that they did, and that they disposed of wild animal bones differently than domestic bones (perhaps they returned them to the forest?). Perhaps hunting took too much time / energy, and diverted too many resources from the farm. Perhaps wild animals were 'unclean' or somehow not fit for consumption by decent humans. We could look to anthropology for a long list of possible reasons why someone wouldn't eat a wild animal, but at the end of the day we don't have an answer that we can prove. All we can say with certainty is that there is no evidence from farming settlements that people ate venison in the 5th and 6th centuries, and no evidence that it was eaten outside special occasions that involved the whole community after the 7th. And by the end of the Anglo-Saxon period, only the trash heaps of the richest people show evidence of a venison diet.

Further reading: Naomi Sykes published an excellent survey of the evidence in 2010, and most of my answer is paraphrased from her. Newer excavations (such as Lyminge, Kent) show patterns consistent with Sykes' argument.