Just curious is there a reason why it's random?
As a thought could it be possible to have ranked in a specific order such as sup, jg, top, ad, mid or any actual combination. I'm not saying it's the greatest idea but sometimes having to pick mid first is rough especially if the team doesn't want to swap in benefit. I understand the pro's and con's of this, just looking for other peoples opinions.
I'd assume it is to prevent one team from always having the opportunity to counter-pick in specific lane.
If top-lane on blue-side always picked before red-side, it would be rather annoying.
I've heard it said that (Now this is NOT a Rioter's comment on it, just a fellow player) being first pick will try to induce players to work with teammates. Such as, a personal experience of mine is...
My most recent game I was the Top-Laner for my team. I was First Pick. As in, first pick after banning phase was done. Thus, I did not want to be counterpicked for grabbing the person I hovered so early on (It was 
just so you know) so I asked my teammates if there were any champs they wanted in particular so I could lock them in for them.
So that MIGHT be an indication that they'd LIKE people to be more congenial.
its ALWAYS been random??
and even saying "pick order" being up bad memories of idiots who would think "first pick" gave them the right to screw the rest of the team over, instead of WORKING as a team to get EVERY player a role they can at least PLAY.
i get your point but if they were to go to this set up as you suggest then i know and every one knows that the LUX 1st pick is support and that maokai 2nd pick is jungle and so on... and it would be easier some might argue to pick around a team comp if you know what champ is going to what role
i understand well you can swap and trade picks.... you can still do that
i understand not every one has all champs....
but as of now if im 1st pick and i pick a flex champ like maokai who can go top,jng, support ( sometimes mid very underrated vs ap melee assassin like kat ) the enemy has to guess where im going what role/lane and pick accordingly and if i am TOP and get a hard counter picked afterwards i can then say HEY DOES ANY ONE WANNA TOP ? and i swap roles with out swaping champs like maybe the support is all like yea ill go top then they counter pick the enemy champ that countered mine and now im still my champ and not gonna get pubstomped by being countered pick
but once again it goes back to people being a team communication and working together which we all know has flaws
just my thoughts
