Our "royalty" are chosen by a tournament of single (i.e., one-on-one) combat. Either gender may fight, but (in Crown Tourney) must have a "consort" (of the opposite gender) to fight for. Thus, the only qualifications are that you be a member in good standing, and either good at combat or charming enough to convince someone who is, to name you as their consort. Note the total lack of need to be a good leader, or even a nice person, let alone well-rounded, or at all aware of how the SCA functions, the laws of the Kingdom (except some pertaining directly to Crown Tourney), etc.
We've been lucky enough not to have had many total jerks on the throne yet, but the system certainly allows it... and skews the royalty exclusively in favor of fighters and their favored, and quite strongly in favor of those who have concentrated so much on their fighting skills, that they may well have neglected other facets of the SCA. It also gives the highly mistaken and (IMHO) very bad impression, that combat is the be-all and end-all of the SCA, to the denigration of dance, music, heraldry, cuisine, garbing, etc. -- even the very craftsmen who make the armor that Crown Tourneys are fought in. (Yes, some fighters make their own armor -- but even so, winning Crown gives them recognition solely for their combat skills, not for their armoring skills.) Furthermore, it specificially places the emphasis on "heavy list" combat, as opposed to rapier or archery, and even more specifically on single combat, completely ignoring all aspects of military tactics and strategy. Never mind the complete ignoring of the courtesy and service that many consider the core values of The Dream, tho I suppose that carrying out the royal duties is service.
As you may have guessed from the above, in mine own 'umble (?) opinion, and that of many others, it is high time for a change. People have been griping about this for literally decades. There are plenty of workable alternatives.
Changing Crown Tourney from single combat to melee (i.e., whoever can command the best army wins), would at least require leadership -- and the "leader" need not even be a fighter himself, but can direct from the sidelines. (It would also be much more historically accurate!) Even just allowing a couple to hire a champion, rather than requiring that one of them fight directly, would greatly expand the pool of potential monarchs. It would also allow those fighters who don't have the time (or desire) to actually be monarchs themselves, to fight in Crown Tourney.
Tournaments of games and other mental challenges could be included, perhaps an Arts and Sciences competition, along with contests of dance, impromptu versifying, and other gentle arts. The possibilities are endless, and would serve to widen interest in the activities that the SCA has to offer other than combat.
I envision something like a set of tourneys, in which points are added up, tallied along with additional points for how long one has been in the SCA, previous service done (anything from helping set up tables at events to holding high office), etc. For instance, perhaps up to 10 points could be awarded for each of the following, with the winner being the contestant with the highest total :
("A tourney of one" means that there will be tourneys of several, but each prospective monarch may enter only one.)
A prospective monarch may therefore "sit out" a tourney, if he (or she!) is rather certain of making up the points in another one -- but the well-rounded will, in general, hold an advantage.
To help guard against jerks, perhaps up to 10 points could be deducted, if so voted (via secret ballot) by whatever group may be polled, such as either the landed barons of the kingdom, all who attend Crown but are not prospective monarchs/consorts, etc. Of course, this would probably lead to all sorts of political infighting and backstabbing, but hey, at least that's period! B-)
Making it a set of tourneys would also have the effect of making it a much larger event. This is, IMHO, quite befitting the choosing of a new monarch! In reality, this would be one of the most important things in a gentleperson's life, not just another event.