I understand that Edward, being a seemingly fanatical Protestant from the little we can gather of his short life and reign, wanted to secure a Protestant succession and was determined that his piously Catholic sister would not succeed the throne following what looked to be his oncoming death. What I can't wrap my head around is why he, amusingly following in his father's footsteps with a desperate lack of a male heir, chose to pass over Elizabeth in favour of the more obscure Lady Jane Grey, whose claim to the throne rested on being the granddaughter of Mary Tudor, daughter of Henry VII.
The most typical answer seems to rest on Elizabeth's bastardry, as he couldn't in good conscience pass his throne onto someone who had no right to it. That has never made much sense to me as if a bastard cannot inherit, why was Edward so frantic to push Jane - and her male heirs to be specific - so much when neither of his illegitimate sisters could inherit either? The answer, of course, is that his father had rather inconveniently for him returned Mary and Elizabeth to the line of succession. So if the answer as to why he was refusing to pass the throne onto Mary was a legal argument, it was clear Edward didn't give a damn about the law as he was clearly willing to overturn Acts of Parliament and his own father's will explicitly stating the throne should pass to them if Edward should die with no heirs. If all Edward wanted was to prevent Mary, why was Elizabeth not the natural choice for him? And there were plenty of ways he could ad hoc this decision with any line of legalistic mumbo jumbo that could be far more rationally justified than a relatively unknown cousin.
An alternative answer in cases such as these might be personal animosity, that while they might have shared the same religion, Edward didn't like Elizabeth much either, as all the evidence indicates he held typically male chauvinist views given how many times he had to revise his Device on account of there being no male heirs to inherit. It's perfectly plausible that Edward didn't trust either sister, but that doesn't appear to be the case. Indeed, as much as royal siblings can have a productive relationship in this period, Edward and Elizabeth got on rather well, embracing each other when they got the news of their father's death. The same is true of Mary, but while the religious question seems to have fractured the relationship between Edward and Mary, the same cannot be said of Elizabeth who remained the quiet, dutiful sister to her brother while he was on the throne. So why then did Edward not choose her?
It has never made sense to me that if Edward wanted to prevent a Catholic heir, he would not choose Elizabeth who had more of a claim to the throne than Jane, matched Edward's religious preferences and Edward got on with and would have had no personal reason to bar from the throne, save the fact that she was a woman but since he was willing to dispense with that standard for Jane's sake, why not Elizabeth's too? The only argument I can rationalise is that Jane was already married so effectively guaranteed to soon produce male heirs unlike Elizabeth but since she would have come to the throne much younger, I think she could have been more successfully pressured into marrying than in our series of events, especially if Edward stipulated as such in his device.
It's entirely possible there is some factor I've overlooked or am simply ignorant to, but the usual answers I get from this topic don't really seem to add up to me as if Edward was willing to break the law to protect Protestantism and place his cousin on the throne, why not his sister instead?
There are a few pieces of the puzzle that your breakdown is overlooking. Namely, the influence of John Dudley, Duke of Northumberland.
In the Third Act of Succession, Edward's father, Henry VIII restored Mary and Elizabeth to the line of succession. Henry also created a group of nobles to oversee Edward after his demise in order to ensure that his rule proceeded smoothly and the line of succession wasn't threatened. Henry himself had been obsessed with leaving England a suitable mail heir but recognized a very real rational limitation: there were precious few male Protestant heirs with connections to the throne. Edward VI recognized this limitation even before his falling ill, so nominating a female candidate was more of a necessity than a wish. But it was also something that his father's legal will meant to protect as a backup in case the need ought to arise.
The group of nobles that Henry VIII entrusted to oversee Edward VI did not act in the way Henry intended. John Dudley, Duke of Northumberland, became Edward's chief custodian in 1550 after replacing the previous Lord Protector, Edward Seymour, Duke of Somerset, in a coup. Dudley was noted for being more competent than Seymour but also sometimes heavy-handed, especially when it came to religion. Dudley was more of a radical Protestant than Somerset was and became a close associate with Thomas Cranmer, another Protestant reformer. Edward VI supported this type of Protestantism, breaking from his father's more traditional approach to accepting further radical transformations to the English church. As Lord Protector, Dudley had a very close relationship with the king and when Edward fell ill he oversaw the writing of the King's device that designated Jane Grey as the heir. Jane Grey being a distant cousin, and also Dudley's daughter-in-law.
The two main arguments for Jane Grey derive from this relationship. Dudley oversaw the writing of Edward's device, was it mere happenstance that said device which defied Henry's wishes for inheritance came at the cost of selecting a candidate who was married to the Lord Protector's son? Dudley was in effect putting his own family on the throne. It looked like an egregious power play. But on a more ground-level Jane Grey appears to have been more devoted to the Protestant cause than Elizabeth. She was educated in the same kind of Protestantism that Edward VI and Dudley shared. If Edward and Dudley were seeking a candidate who would continue England's religious transformation then Jane Grey looked like a suitable candidate. Dudley could personally vouch for her on this. What made Elizabeth unviable came as a consequence of this relationship between Dudley and Edward VI. Elizabeth was associated with the old regime, after all. Her father-in-law had been the brother of Seymour. Further, she was not raised to share the same Protestant beliefs that Dudley and Edward VI held.
Dudley's personal influence on the king, the shared radical Protestant beliefs held, and Elizabeth's political background made her undesirable as a primary successor.
Sources:
David Loades, The Tudor Queens of England
John Edwards, Mary I: England's Catholic Queen
John Guy, The Tudors: A Very Short Introduction