What if Scotland's dynastic uncertainties had begun prior to Edward I ascending to the throne of England?

About the middle of Edward I of England's long, successful and ambitious reign as King of England, the then independent Kingdom of Scotland began experiencing serious conflicts and instability regarding the proper dynastic succession to the throne of Scotland. It was unclear who had the best claim to the throne -- there were actually 13 credible competitors for the throne of Scotland. https://en.wikipedia.org/wiki/Competitors_for_the_Crown_of_Scotland To avoid civil war, the Scots Lords actually formally requested Edward I of England to intervene as an "honest broker", to decide the issue. At the time, England and Scotland had a rather amicable relationship, and the highly successful King of England seemed like the right person to decide the probable best course for the monarchy of Scotland. Also, there was some tradition of English overlordship in Scotland, although the issue here was somewhat equivocal. Edward decided to take advantage of the situation to advance English claims to overlordship of Scotland, and conflicts between himself and the Scottish lords on this issue eventually led to him leading an English invasion of Scotland in 1296. He eventually succeeded, with some difficulty, in establishing effective English control over most of Scotland, and by the time of his death in 1307, Scotland was largely an English dominion. However, Edward's successor, Edward II, was a weak leader, and Scotland gradually gained ground against the English, obtaining full independence within a couple of decades. So, let's suppose that these problems of dynastic succession had begun some years prior to Edward I ascending to the throne of England, in 1272. How might this change history? Given such a golden opportunity right at the start of his reign for imperial conquest, there seems little question that the young Edward would have jumped at the chance. So, we might see an invasion of Scotland a generation earlier than OTL. And, Edward I being a brilliant soldier, this invasion is likely to be successful. And, Edward is likely to be able to maintain control of Scotland till the end of his reign some thirty-five years later. This is quite a long time to retain control of a country. Likely, it would be long enough to establish some English traditions and colonization even in the Scottish highlands, and throughout the nation of Scotland as a whole. So, a subsequent Scottish revolt against English overlordship becomes much less likely, and long term control of Scotland by England almost inevitable. Given a United Kingdom of Scotland and England from the thirteenth century, a War of the Roses becomes less likely, and the hundred years war in France, if it occurs at all, seems rather more likely to turn to England's favor. England will have more resources, more manpower, and less competition. England may well retain control of much of France indefinitely. Thus an English empire across all of Britain and much of France has the potential to, perhaps, turn the whole of Western Europe English, perhaps by the sixteenth century. Thoughts?