Crown Tourney should remain a martial tournament, rather than an Election or an Arts & Sciences competition.
First and foremost, it's the mystique. "Our king and queen are chosen by combat." That's so thoroughly unmodern that it immediately identifies this group as different. Not even professional sports teams choose their captains in head-to-head competition.
The SCA calls it's major events Wars. If a King or Queen isn't possessed of the prowess to lead on the field of battle, something is lost. Generals can be appointed, any fighter who won Crown by an alternate method could lead, but something is lost.
Tradition. Sometimes tradition is worth maintaining. It is true not all traditions are worth maintaining. But the selection of a king and queen by combat is literally the bedrock of the SCA. If that is changed, the very substance of the organization is changed, and not in a good way, in my opinion.
Other methods don't inspire the same gravitas and excitement, don't create the same suspension of disbelief that let's us all 'believe' in our royalty. Please don't get me wrong, I regularly stand in awe of the artistry and craftsmanship on display in the SCA. But a judged A&S tourney isn't going to make me 'believe' the winner is royalty. Neither a judged bardic competition. I'm too thoroughly raised on the Le Morte de Arthur and Lord of the Rings. Thrones are won and maintained by the sword.
And the parameters for a judged A&S or bardic tourney are never going to be the inclusive fit one might hope for. Time limits for construction, research, etc. favor those with greater disposable income and leisure time. Quality of finished product favors the same group. Judges' bias, whether malicious or just lack of interest in subject matter, will always be a nagging concern or accusation. And without crushingly exhaustive documentation, how does one prove this thread was handspun by oneself rather than ordered from Etsy?
The best fighter does not equal the best administrator or even the best ruler, this is very true. The king is not selected to balance the books. Nor will any mortal be a perfect ruler, and to that I say, the poor rulers make us so much more enthusiastic about the good ones.
There are excellent arguments to be made for redesignation of particular functions to elected committees, because we do have modern concerns with internal administration and rule enforcement.
Royalty should be in charge of the fun stuff. The serious stuff with potential legal ramifications should be handled by persons with the proper modernly recognized credentials.
Is there a place for champions? That's already literally implemented, every fighter has a consort. This is a social club, is it not? Make friends, have discussions and long talks, and find a fighter to partner with. "Help me achieve the throne, we will have so much fun, and I will clothe you in masterpieces (or write magnificent ballads, furnish your camp, etc)." Being perfectly honest, if that's not the sort of offer you can make, what are your odds of winning an A&S Crown tourney?
Is there room for more 'leadership' positions chosen by A&S tourney, etc? Definitely. I want to see the Arch-Laurel heading up Kingdom-wide projects. I want to see the Eminent Pelican and their minions aggressively feeding the army next door.
I have questions for those who want to change the format of royalty selection. Why do you want to be royalty? Is it only because you want the title? Are you trying to win the SCA? What can you do as royalty that you can't do otherwise?
What aspect of royalty is lacking that can only be satisfied by shifting the selection pool to artisans?
For myself, I enter Crown Tourney with my consort because I believe I can play into the mythos - I can lead the army and laugh with the bards and extol the great works of artisans and notice the people who help quietly in the background. Perhaps this is hubris. But I'm not in it primarily for myself. I'm in it to see what I can do for the kingdom, if I can play the role well enough that when you squint and stand back a few yards, you might wonder for a moment if you've fallen completely out of the modern world.
Making more categories of competition doesn't always make the competition better. What good is there in saying "I won" when you also have to say "after they changed the rules to favor me."
Because I must,
Ren Jie
"Lá melan i macil calina an laicasserya,
lá i pilin an lintasserya,
lá i ohtar an erya alcar.
Melan eressë sa i entë tirir: i Minassë Núatanion"
"I do not love the bright sword for its sharpness,
nor the arrow for its swiftness,
nor the warrior for his glory.
I love only that which they defend: the city of the Men of Númenor."