That'd be crap though, that wouldn't be exciting that'd just be; have some wets ready, oh there go the sprinklers, *every car pits*, 23 laps time when the track dries everyone comes back in for slicks.
Rain introduces excitement because you have one team who reckon the rain will come back and so keep their cars out on the wet tyres whilst another team reckon its about to stop and so take the bold move of putting inters on and so on. Most importantly though, it's based on data and is a prediction. Some teams will see rainfall but keep cars out on slicks because they reckon its a 2 minute shower and they'd lose more time doing 2 pitstops than lapping 5s slower for 3 laps and stuff like that.
A fixed 20 minute sprinkle would recreate none of that and neither would having random length/strength sprinkles either as then it'd all be total guesswork and you'd get drivers suffering just because they happened to guess wrong, it'd be little more than straight 50/50 gambling.