Why did England *apparently* function much better between 1066 and 1330 when the King was attending matters abroad?

I don't think England functioned better during that time. Henry I saw significant pushback on his right to rule by his brother Robert, understandably as he was not next in line, and while he easily kept his throne his brother did have support in England. The Anarchy was the worst period of post-Norman English history until the War of the Roses. It was a period of on and off civil war and the people might not know who their ruler was at any given moment. It lasted for nineteen years. This was a deeply unsettled time.

Under Henry II, who is certainly regarded as one of the strongest kings, faced a serious revolt led by his sons and wife in 1175. While much of that happened on the continent, there was significant fighting in England with Northern lords against the king. Spending time abroad didn't prevent revolt at home. While Richard I was on crusade and in captivity John led a rebellion against him that did gather some support although Richard was a very popular king.

In the matter of John, there were a lot of things going on.

First, the problems within England had begun before he took the throne and lost their French possessions. England was a fairly prosperous realm. But under Richard it had been squeezed tightly for revenue to fund his very expensive crusade. At that time the other lands were taxed as well but there was never the same level of efficiency and most of the lands didn't have the same level of wealth. Basically, England was always squeezed the tightest when money was needed. Then when Richard was being held hostage Eleanor (and John) had to squeeze even harder to come up with the enormous ransom. So there was already a backdrop of discontent. It may never have lead anywhere but problems didn't arise suddenly under John.

Second, John's campaigns to fight off the French in the French territories were expensive and they failed. Then John had to find more money to try to reclaim the territory, which also proved expensive. By now England was the only place to try to raise the money. (More on this in a moment.)

Third, during this period kingship was still a personal thing. Henry II was a good administrator and legal reformer. But to his nobles he was primarily someone to lead them into battle (and he was very good at that). Modern scholars have found things to admire about John especially in the legal and administrative realm, which are important. But men still wanted a leader who they could be proud to follow in war. That was a primary, ancient function of kingship. From childhood on John was never someone who commanded a lot of respect compared to his brothers or father. But that's probably the burden of youngest sons. However, when as king you lose half of your territory and efforts to reclaim them are total failures that's very difficult to come back from.

Fourth, in order to pay for new campaigns on the continent John resorted to a lot of very unpopular tactics. He exploited the use of fines, payment for charters, and fees due to him. He overused scutage, which was a way for nobles to pay money instead of providing service in battle, by levying it even when there was no military campaign at hand. He also exploited the ability to charge for making valuable appointments (like sheriff). Many of these things were actions done by Richard as well and just expanded under John. But Richard was a successful military commander. Also, there is some evidence that there were some times of famine during this period.

Fifth, John got into a lengthy battle with the Pope which led to England being under interdict for some time. While under interdict you couldn't go to mass, be buried in a churchyard, get married in a church. For a religious society it was painful. But it was profitable for John because incomes that would usually go to the Church went to him. This led to a build up of resentment toward him and a general feeling that he was deeply unpious and untrustworthy.

All of this (and more) lead to the revolt of the barons. The king was for various reasons not someone they held in high esteem and they felt he was taking advantage of his position as their liege lord and exploiting them. It was in some ways not something that was entirely John's fault, the problem of abusing feudal rights to raise money began under Richard, but it was a problem he exacerbated. Kingship was a personal thing at the time and they didn't like his person. It wasn't why it happened but it didn't help.

Henry III inherited many of these problems and they simmered while he was in his minority but didn't die away. He exacerbated them but they weren't issues that arose merely due to proximity.

So, no, I don't believe that John or Henry (or Edward II) was a case of familiarity breeding contempt. There was turmoil under the earlier Normans and Plantagenets.

I think under Edward I and Edward III there was stability in England that even very strong, capable kings like Henry II might envy. (I know your time period includes up to 1330 but that is so long after the loss of the French lands.)