I'm looking at the Wikipedia page for the list of English and British monarchs, and it's kinda weird that only Victoria and Elizabeth II appear painted with their respective crowns. Pretty much none of the kings have their crowns.
As a matter of fact, it seems like most kings aren't painted with crowns at all, including those of France. I look up George III on google and almost none of his images feature a crown, just ermine and copious amounts of tassels. Why is that?
Hey! I'm an art historian with a research specialisation in early modern portraiture and I also work in a museum about royal history. While it's hard to bring one clear answer to this I might be able to provide some context and further knowledge.
But when investigating if and why crowns were or were not worn in royal depictions, and if female monarchs were more or less likely to wear them, we can consider the following.
We must take into consideration the functions of portaiture and the function of crowns as well as consider different uses and needs of both through history. While part of the symbolism and regalia of monarchy, a monarch doesn't always wear their crown. In fact, the crown as an item is historically highly linked to the coronation, which only happens once. Depictions that feature them are often coronation portraits and the likes where the monarch is presented with parts or all of their regalia (apples, keys, swords, scepter, etc). This regalia is meant to embody and embue the different parts of divinely appointed kingly power with which the monarch in question has been bestowed. But with how the coronation and anointing works the monarch themselves is not ever actually required to display the items to be considered as such. The regalia, of which the crowns always are part, are thus symbols of state and monarchy but do not need to be physically worn by the monarch to serve that function. That way it often suffices to just have them on a pillow next to you in a portrait which is perhaps the most common through history. The fact that they are included in a portrait at all is indicative enough of monarchy as nobody else but the monarch is allowed to even associate themselves with their symbolism and definitely doesn't have access to them in a way that allows them to be painted.
Now to consider some of our biases in depictions of the jewelry itself. There's a difference between state crowns and tiaras or diadems. The very fact that women have traditionally worn tiara and crown type jewelry as, well... jewelry while men have not means that there is a whole range of tiaras and diadems in royal collections that female monarchs wear as top-woman-in-case rather than as monarch-in-case. The best example of this for QEII is The Girls Of Great Britain tiara which has come to be closely associated with her and was featured in a lot of depictions but has no state function and is technically just a piece of jewelry that she owned and liked. As for actual royal crowns the monarch only officially wears the St. Edwards crown once for their coronation ceremony (and then possibly again in private for painting of coronation portraits, but it's extremely heavy so I doubt anyone wears it more than they have to.) The Imperial State Crown (that rested on the coffin of QEII) is then used once every year for the opening of Parliament. This means that many depictions of Queen Victoria and QEII may be depictions of them in other pieces of jewelry that are not state crowns, that they wear simply because it is what is customary for royal women to wear and that this skews our perception of their image as more often including a crown type item on their heads. In terms of fashion history royal women didn't really wear tiaras before the early 19th century, which means that most of the time between then and now has seen a female ruler on the British throne. This includes what I would deem the golden age of tiara making and wearing of around 1870-90.
Now onto the function and method of portraiture: Queen Victoria and QEII both lived in a time where the meaning of portraiture underwent an enormous change. Photography became an increasingly widespread medium during the first few decades of Victoria's reign, and digital photography became widespread during the reign of Elizabeth. This increased availability in practice meant that the role and conventions of royal imagery had to change when society was suddenly flooded with depictions. Through history portraiture (for state functions generally through the medium of oil painting) has been a privilege of the few, meaning that the pictures that were created were more compositionally curated and staged to include certain markers of royalty. Once photography entered the stage it became pretty clear that flying babies, naval warfare backgrounds or mysterious levitating drapery didn't work quite as well for an actual snapshot of an actual human being. This may have placed larger emphasis on the crown as one of the royal symbols that still translated well into this new medium.
With the new medium of photography and influx of imagery also came different means of visually linking certain persons to their functions. QEII is one of the most depicted people in history and the way in which she has been repeatedly painted and photographed has been done in a completely different way to how monarchs would have been depicted previously. Back in the 16th century QEI had Nicholas Hilliard and a select number of other painters standardise her "mask of youth" which was then repeated and copied - she definitely did not sit for every portrait we have of her. In the 19th century Queen Victoria had Winterhalter as her court painter and preferred having him paint her portraits for several years. For portraits like the George III you mention a court painter (In this case Romney) was appointed to produce an official image of the king and his studio would then crank out copies of that portrait to go around. All these examples mean that the picture of the monarch is copied and spread without their own participation in having to sit for every portrait as the point isn't to have a new creative expression but another copy of a chosen image. In our time royal portraiture works a bit differently. Queen Elizabeth II didn't keep a court painter responsible for her outward imagery but was instead likely approached by the artists or whoever commissioned a portrait by an artist, who then came to her and did their thing in their style. This means that QEII sat for countless artists and was depicted by a whole range of artists with different ideas and styles and visions and methods in a post-modern time where we see images differently and adhere to multiple traditions of style and expression as opposed to just one of "this is the official depiction". The official function of the crown itself depicting and conveying monarchy, with crown imagery and all, is now done by the official photography - think of the official portraits in embassies all around the world.
As for the standard portrait of George III you mention - when there is question of a monarch choosing the style of their standard portrait to be copied and distributed it would have been a combination of the style of the artist, the preference and desire of the monarch as an individual, as well as and the conventions of portraiture and fashion at the time. George III was a pretty laid back guy and not really one for enormous shows of magnificence and it shows in how he chose to be depicted. Hanoverian kings also hadn't taken a huge interest in the arts. Charles II, in contrast, lived lavishly and extravagantly, and his personal preference definitely shows in his famous full-on seated portrait, where he does wear a crown (and a massive one, too!). He also had a lot to prove in contrasting the return of kingship and extravagance with the austerity of the Cromwell years. He had literally been welcomed back from exile to bring monarchy de luxe back into the game - and the style and manner in which he depicted himself needed to help him accomplish that.
As you can see it's a very complex topic where our own biases, the needs of monarchy, the monarch themselves, technical developments, cultural trends, and fashion conventions may all play a part!