There are several factors here, and this is, in fact, a very complicated question as the answer changes over the decades of the 15th century. So we are going to need a lot of background before we can answer this question:
First of all was the Burgundian rule of Flanders. Flanders was far more wealthy and urbanized than Burgundy, and quickly became as, then more important to the Burgundian dukes for this reason, as well as because it served as a base for expansion into the Low Countries. Flanders had a long historical relationship with England, as Flemish cloth makers were dependent on English wool. Therefore, Flanders could never be as anti-French as the rest of France. Remember that Flanders was legally subject to the French crown and therefore technically part of France, but effectively autonomous during this period, which led its counts to steer a course between England and France during the Hundred Years War. Once the Valois Dukes of Burgundy -- who were, of course, French royal princes -- inherited Flanders, they immediately fell into this pattern, as it was dictated by geopolitics rather than personal desires. Phillip the Bold and then his son John the Fearless both had to steer a course between their role as leading French princes and counts of Flanders.
Now, we must turn to the Armagnac-Burgundian civil war which led to the final stages of the Hundred Years War and the Anglo-Burgundian alliance sealed with the Treaty of Troyes. There's a lot to be said here, but in short, there was a vacuum in power because Charles VI's madness made him incapable of ruling. This led to competition between Louis, Duke of Orleans, who as the king's brother was the natural regent and used this position, and John the Fearless, the king's first cousin who wanted to assume his father's (Charles VI's uncle) role at the heart of French government. In 1407, John the Fearless had his cousin the Duke of Orleans assinated and made a bid to take control of France. However, he was unsucessful and his bid for total power led to a war with the Orleanist princes, now called Armagnacs, who were basically most of the great French princes -- the Dukes of Orleans (Louis' son), Berry (Charles VI's uncle, last of his generation to survive), Bourbon (another royal prince), and the Count of Armagnac. This civil war literall tore France apart, as each of these princes controlled their own regions as the fought for power of the French government (which controlled much of France directly). Ultmately a failed proletarian revolution in Paris (the Cabochian revolution) drove John the Fearless from Paris and back to his own domains in Flanders and Burgundy. (On a side note, John was the leader of his own coaltion of princes, mostly his kinsmen, the Duke of Brabant and the Counts of Nevers and Hainault).
Here's where Henry V enters. Seeing the chaos in France, Henry of Lancaster saw a golden opportunity to win back some of England's lost territories, at primarily the enlarged Duchy of Aquitaine created in 1360, then lost little more than a a decade later. Henry went on to win an incredible victory at Agincourt in 1415, then counquer Normandy between 1417 and 1419. Now, there are three factions vying for control of France: the Burgundians in the East, controlling a the King and a puppet government, the Armagnacs in the center and south, who controlled the Dauphin who was heir to the throne, and the English in Normandy. In 1419, the English began to invade the Île de France, the heart of the French government and were dangerosly close to Paris, currently under the control of the Armagnacs but surrounded by Burgundian garrison. Even now, the hatreds generated by the civil prevented the Argmagnacs and the Burgundians from allying with each other! However, John the Fearless, afraid of what could happen if the English were left unchecked, sought an alliance with Armagncs, now led by the Dauphin Charles and called the Dauphinists (so to be clear, the anti-Burgundian party has been called the Orleanists, Argmagnacs and Dauphinists at different points).
Famously, however, John the Fearless was killed by partisans of the Dauphin during a meeting a Montereau to discuss an alliance. This was a grave error, as the Burgundian domains and leadership of the Burgundian party fell to Phillip the Good, John the Fearless' son. Phillip the Good was forced to ally with the English against the Dauhin (Charles VI is still king even after all this time!). Phillip did not have the stength to fight both at the same time. Therefore, at the Treaty of Troyes, Henry V was recognized as the heir to the French throne when Charles VI, and the English and Burgundians made an alliance to defeat the Armagnacs. Phillip the Good was still at this point a French prince more than an outside power, and took the least bad option to save himself, gain some territorial concessions and have his autonomy guaranteed, and have a chance to defeat the men who killed his father.
Now, instead of going into the details of the Hundred Years war between 1420 and 1453, I'll summarize the Burgundian developments:
Phillip the Good used his position to reinforce his autonomy and directly take control of the Low Countries -- that is to say, Brabant, Hainault, Holland and Luxembourg. He abandoned the English alliance at the Treaty of Arras in 1435, once they lost the upper hand, and became effectively a neutral party between the English and French, which allowed the the French to defeat the English and expel them from France. This was really the period where the Dukes of Burgundy went from French princes to effectively indepent princes, with resources comparable to most Western European monarchies at the time such as England or Castile. After his victory, Chares VII (the Dauphin mentioned above) found himself faced with an extremely large and wealthy Burgundian state to his east.
At Phillip the Good's death in 1467, Charles the Bold sought to preserve his de facto indepence from the French crown. He in fact did seek his own crown, and there was a real possibility of this happening. In 1473, Charles the Bold met with Holy Roman Emperor Fredick III to discuss the erection of his imperial territories -- the territories he held from the Emperor rather the the King of France, namely the County of Burgundy, Brabant, Holland, Hainault, Luxembourg and recently Guelders -- into a kingdom. This kingdom would have been still subject to the Emperor but would have been an enourmous increase in prestige. However, this did not happen and Charles the Bold was defeated and killed four years later at the Battle of Nancy in 1477, his dreams of a kingdom never quite realized. His territories were inherited by his daughter Mary of Burgundy and passed to the Habsburgs, save for Picardy and the Duchy of Burgundy proper, which were absorbed by the French crow.
Wow, I did not set out to write that much!! This is a very complicated question that I guess deserves an equally complicated answer, especially since different decades had different aswers to your question I have brushed through about 75 years of history here so let me know if you have an questions.
Sources:
Follow-up questions:
what would have been the advantages of becoming an independent kingdom?
would becoming a kingdom require the pope's approval? If so, how was Burgundy planning on securing this?