The short answer is that Flodden was absolutely a fluke. After the battle the English had no reason to suspect that the Billhook was at all capable of defeating the pike in a conventional military engagement. The circumstances at Flodden were extremely unique, and it is hard to imagine that more unfavorable conditions for the use of massed pike formations could have been deliberately engineered.

The first thing worth saying about Flodden is that the Scots absolutely should have won the battle according to the prevailing military thought of the time. They had an ostensibly modern army with a superb artillery train, pike armed infantry emulating the latest continental style, and were lead by a forward facing Renaissance king supported by a cadre of French military experts. By contrast, the English army would not have necessarily looked out of place following Edward III to war a hundred and fifty years earlier. It was armed with outdated weaponry (bills and bows, which remained prevalent in England for quite some time despite the fact they had long since disappeared from continental warfare) and while the English artillery train was entirely modern in design, it represented the dregs of the English arsenal as all the big guns were in France with Henry VIII. However, it is worth noting that the Scottish army wasn’t quite as modern, and the English army not quite as backwards facing as they may have seemed at first glance.

By this period pikes had pretty definitively proven themselves as the preeminent formation weapon on European battlefields. Although by no means invincible, formations of men armed with pikes had been shown to be reliably capable of defeating almost any other combination of arms on the battlefield. However, in order to remain effective pikes had to be used in a close order formation. Unit cohesion was absolutely essential for the successful use of pikes in combat. When and if unit cohesion broke down you were left with a bunch of individual soldiers holding incredibly unwieldy 18 foot poles. In very simple terms, the pike was absolutely useless as an individual combat weapon, which meant that outside of close formations pike armed soldiers would be largely at the mercy of soldiers equipped with more dynamic weapons.

At flodden the Scottish pike formations broke apart early in the battle. The Scots were forced by circumstance, and good generalship on the part of the English, to March down Braxton Hill to face the English. Unknown to all sides, at the base of Braxton Hill was a marsh caused by water run off from the surrounding hills. By all accounts the ground looked firm and entirely passable from a distance, but once you set foot on it it became a treacherous, sucking morass. The part of the Scottish line that passed through the marshy terrain broke apart as men stumbled on the treacherous footing. The unit cohesion that was critical for the success of pike armed infantry was completely lost. Once the Scottish pike formations broke apart they were relatively easy prey for more nimble Englishmen armed with bills and poleaxes. It is worth mentioning that in contrast to the pike, which was useless in loose formations or individual combats, the Elizabethan fight master George Silver considered the English bill to be a weapon without peer for individual combats.

It is important to recognize that the failure of the Scottish pikes has very little to do with them meeting a “superior” English weapon. While the section of the Scottish line that crossed the boggy terrain was relatively easily defeated by the English soldiers, this was almost entirely due to the total collapse of the tight formations required for effective use of the pike. Elsewhere along the line the Scottish pikes performed quite well. The English right flank had to be rescued from near collapse by the English cavalry under Lord Dacre, which charged decisively into the flank of the Scottish formation. Under more normal battlefield conditions there is no reason at all to suppose that the bill would have had any advantage at all over the pike. Contemporaries seem to have recognized this as well. As you noted, the English would steadily replace their traditional combination of “bills and bows” with more contemporary pikes and firearms. In the wake of the battle the Scottish government banned a weapon called the Jedburgh Stave (a sort of Scottish naginata) specifically due to its poor performance in the battle of Flodden, but make no mention at all of banning pikes.

The age of the Pike would continue for the next few centuries while the bill would be steadily relegated to the dust of history. Flodden was a momentary blip where good generalship and bad terrain counted for more than any kind of superior weaponry.

I hope that answers your question!

Sources:

Barr, Niall. Flodden 1513: The Scottish Invasion of Henry VIII's England. London: Tempus, 2001.

Grummitt, David. “Flodden 1513: Re-Examining British Warfare at the End of the Middle Ages.” Journal of Military History 82, no. 1 (January 2018): 9–28.

Phillips, Gervase. "In the Shadow of Flodden: Tactics, Technology and Scottish Military Effectiveness, 1513-1550." The Scottish Historical Review 77, no. 204 (1998): 162-82.

Phillips, Gervase. “England; Scotland and the European Military Revolution, 1480-1560” The Richard III Symposium (July, 2013)