I'm at work so this is from memory: Henry II did homage and swore fealty to the king of France for his lands in France, and later in his reign his sons did so too.

"Simply declare independence ... destroy any legal avenues they had"? A unilateral declaration of independence (a defiance) had no legal force then any more than today. Or at least the kings of France would have said so, so issuing a defiance would have simply been a permanent declaration of war.

More fundamentally, Henry II wouldn't have. It was fundamental to his character that he insisted on insisting on and enforcing every bit of patrimony and right that he believed he had, regardless of whether it was questionable or useless. This rigidity got him into trouble early in his reign; for example, insisting on the letter of the customs of Henry I against the church when canon law had moved on (the whole Becket thing, the Constitutions of Clarendon).

But he also conceded to all others their rights against him: lands, franchises, et cetera. His new writs (novel disseisin, mort d'ancestor, et al) were fast alternatives to cumbersome customary procedures, or could be seen as the king carrying out his responsibility to see that justice was done. (With a few counter-examples: claimant Henry conceding Northumberland &c to the king of Scots, but later king Henry saying that he could not accede to that. But even that can be seen as consistent: Northumberland was part of the patrimony of England and it would be wrong to alienate it. You could similarly argue his insistance on increasing the knights' fees of fiefs, and note that he backed down against the church.)

Henry II could have hammered Louis VII into the dust, and he would have had good justification at times. He held back. For example, the pacification at the end of the great rebellion was basically status quo ante bellum. (Except for the rebels' castles, but even there, Henry had older English law on his side, where William the Conqueror declared that all castles were the king's to control.)