I actually think random would be better - a set order seems like it would almost exclusively benefit balance/stall teams that can stall until the effect that benefits them shows up, while heavy offense wouldn't be much better off than if they were randomized. I do see how random could be a problem, but I think a lot of the issues with randomness could be alleviated by having the random effect start at the end of the turn instead of the beginning; for example: game starts, Trick Room appears before anyone chooses their moves -> Trick Room is in effect throughout turn 1, then at the end of the turn Trick Room ends and hail appears (but doesn't do damage yet) -> hail is in effect turn 2 and damages at the end of the turn, then hail ends and Psychic Terrain goes into effect. That way you can switch out to keep the hail damage you know is coming from messing with your sash, or use Protect to wait until Primordial Sea ends before nuking that Ferrothorn, etc.Personally I don't think random would be any fun at all. Imagine trying to nuke a ferrothorn with fire blast and getting completely screwed over by Primordial Sea just coming up out of nowhere. It seems really anti-competitive and it's not possible to really account for any of that during team building, like, 'well this pokemon WOULD be good if Trick Room happens to pop up' except you have no way of knowing that so in practice Trick Room would just be randomly screwing over sweepers and mons who plan to take damage before recovering instead of after etc etc. Your focus sash lead suddenly doesn't get that turn 2 because whoops, hail took effect on turn 1.
Whereas with a predetermined order you can kind of plan for that and decide 'okay on the TR turns I'm going to have these 2 mons to send in because if I just have one it's too easy to predict, mon x is just an overall good mon even outside of trickroom whereas mon y takes advantage of other effects like (sandstorm/hail/whatever)', it adds another element to teambuilding and could actually make for some interesting strategies.
Also, would weather from this mechanic overwrite pre-existing weather set by sunny day/rain dance/etc or drizzle/drought/etc or would it be sort of 'phantom weather' so to speak? Would the weather from the mechanic continue on turns where another non-weather effect follows it? Same questions for terrains.