Well, I wouldn't exactly call it 'low birth'. We're not talking the 'king marrying the pretty peasant girl' fairy-tale scenario here. Elizabeth's father was a not-particularly-important knight called Sir Richard Woodville, and her mother was Jacquetta of Luxembourg, who was at least royal-adjacent (her first husband had been Henry V's younger brother). But yeah, she was a commoner, and no one was delighted about that. Edward's privy council pointed out snottily that 'she was no wife for a prince such as himself, for she was not the daughter of a duke or earl'.

So as you suggest, much of Elizabeth's unpopularity - at least to start with - wasn't personal. But the reason her status was a problem wasn't primarily snobbery. It was because it made her a waste of a very valuable marriage. Not only was she a commoner, but just as importantly, she was English. English kings did not marry English women, because the whole point of a king getting married was to put in place a strategic alliance with another country. Edward's mother Cecily Neville and her nephew Richard (Earl of Warwick) had been negotiating a very politically advantageous marriage with Louis XI's sister-in-law for Edward, who really needed a good alliance with France or some major power because the Wars of the Roses were still going strong and his position was not exactly secure, and who if he absolutely had to marry an Englishwoman could at the very least have picked the daughter of someone powerful who would make a useful ally...and instead he blew the whole thing by going and making a completely useless marriage with some random woman pour ses beaux yeux. His supporters were not happy - especially Warwick, who had put all that work into the French marriage, and who now had to go back to the French and say 'Oops, sorry' and look like an idiot.

Speaking of the Wars of the Roses, we also have the fact that Edward was Edward of York, while Elizabeth was a Lancastrian both by birth and by marriage - the Woodvilles were Lancastrians, and her first husband, Sir John Grey, had been killed fighting on the Lancastrian side. In fact, the best-known version of the Edward/Elizabeth story says that they met because she was petitioning him to give back her late husband's lands, which had been forfeit due to that whole Lancastrian thing. Many of the Yorkists had had family members killed by the Lancastrians. Seeing one of those Lancastrians raised above their own families had to sting.

Both of these factors could go a long way towards explaining why Edward and Elizabeth got married in complete secrecy, which was very much Not The Done Thing: they knew in advance that, regardless of anything about Elizabeth as an individual, this marriage was not going to be popular. (Sidetrack: there's also the possibility that the secrecy was partly because Edward was already precontracted to someone else. Years later, when Edward and Elizabeth's oldest son was about to be crowned king, a bishop came forward to say that he had betrothed Edward to a woman named Eleanor Butler, years before Edward married Elizabeth. If true, this made Edward's marriage to Elizabeth invalid, and their children illegitimate. Parliament either believed the bishop's story, or felt that a grown man with a proven track record on both the military and the political front would make a more stable king than a twelve-year-old, because they declared the boy illegitimate, making his uncle Richard king. We'll never know whether the Eleanor Butler story was true - but it would fit Edward's pattern of 'I shouldn't marry her, but I really want to sleep with her and she won't do it without marriage, what the hell let's go for it and I'll worry about the consequences later.') Edward kept his mouth shut about Elizabeth until the negotiations for his French marriage were advanced enough that he was forced to say 'Uh, gee, you know, about that marriage thing...'

So the marriage was a bad decision politically, and caused a lot of tension and anger among the Yorkists. And the blame for it was put squarely on Elizabeth: the old 'scheming woman ensnares hapless man by using her feminine wiles' trope. Elizabeth was by all accounts stunningly beautiful, so that presumably did affect Edward's eagerness to marry her - but her detractors went a lot further than that. Elizabeth and her mother were accused of having made the marriage happen via witchcraft. This allegation was revived in the Act of Parliament I mentioned above: 'we considre, howe that the seid pretensed Mariage bitwixt the above named King Edward and Elizabeth Grey, was made [...] by Sorcerie and Wichecrafte, committed by the said Elizabeth, and her Moder Jaquett Duchesse of Bedford, as the common opinion of the people, and the publique voice and fame is thorough all this Land.' So Elizabeth's unpopularity gradually shifted from impersonal to personal - a shift driven to a large extent by propaganda from Warwick and other Yorkists.

Which brings us to the 'greedy and ambitious' part of your question. There's no way to know whether Elizabeth married Edward out of love or ambition. One the one hand, being Queen of England is definitely a better proposition than being a dispossessed widow from the losing side of a civil war, especially if you've got two young kids to make futures for. On the other hand, Edward was handsome, intelligent, charming, and notoriously popular with women, so it's not like it would have been surprising for her to genuinely fall for him - and their marriage seems, as far as we can tell, to have been a happy one.

When it comes to her family, things look a little different. Elizabeth and her family were very, very serious about getting as many of them as possible into positions of power and advantageous marriages. Among many other examples, her father was created Lord Rivers and was made Lord High Treasurer; three of her sisters were married off to the heirs of earls, another one was married off to a duke, her brother was married off to Warwick's aunt who was extremely wealthy and powerful and about 45 years older than he was (Warwick really didn't like this one), her first-marriage son was married off to a baroness and given various titles; her brother Anthony was appointed Governor of the crown prince's household...and on and on and on. These are all positions that they would never have had if she hadn't been queen. This can be portrayed as either rampant ambition or family loyalty, depending on how you want to look at it, but either way, once Elizabeth got going, there were Woodvilles everywhere at court.

Edward's supporters - in particular his brothers George and Richard, and Warwick - didn't like that one bit. The Woodvilles were getting their hands on things that Edward's own family and supporters had wanted for themselves - Rivers was replacing Warwick as Edward's most influential advisor, marriages that Yorkists had wanted for their own heirs were being arranged for Woodvilles instead. And bear in mind that the Woodvilles had been Lancastrians. While this had a political function for Edward, sending the message that he wanted to forgive old enemies and build unity, his supporters were about as happy with it as the Democrats would be if Joe Biden suddenly started handing over all the most coveted jobs in his administration to random members of the Ted Cruz family. The Woodvilles were coming closer and closer to running the country. George and Warwick eventually rebelled against Edward and temporarily threw off him off the throne, to a large extent because ENOUGH WITH THE BLOODY WOODVILLES.

(Return to sidetrack: this also plays into the Richard story. After Edward IV died, Richard - who had been named Lord Protector in Edward's will - removed Edward's son from the Woodvilles. This is often presented as 'AHA!! He was plotting to steal the crown!!' but I think it's at least as likely that he just couldn't bloody stand the Woodvilles, and was very aware that leaving the kid with them would be tantamount to putting the entire country in Woodville hands.)

So overall, it's not that Elizabeth's social status was the main reason why she was unpopular, or why she was portrayed as greedy and ambitious. It's more that she started off being unpopular for political reasons, didn't help anything by going all out on her family, and then her social status was used by her detractors to bolster a (not wholly unjustified) picture of her and her family as over-ambitious, grasping schemers.