Of course where it would be applied would need SIGNIFICANT thought.
It would likely cause more problems than not.
If you apply it on 1 item, the best users of the item are the best champions.
Apply it on a new champion, new champion could be #1 OP rather than ~20 mobile champions being top-tier.
You'd basically need to go back and apply it to a solid ~20 existing champions to actually offer the type of answer you're after, and if roots aren't doing it for you already - it also begs the question of how effective such a change would really be.
Also - I don't think mobile champions are a problem specifically due to being mobile.
If you look at the most prevalent tournament picks, having 7 skills or an extreemly flexable skill-set is usually the more relevant thing:
All of these have maintained being solid picks for at least 2 seasons now - which makes the issue more champions with a lot of defense/cc/sustain/utility/range/vision/early power/global pressure/flexability *and *mobility more the problem, not just being mobile as an end-all.
There are a lot of mobile champions that are mediocre and balanced, where anti-mobility would harm them tremendously. Fiora and Riven can't cast Lunge, Blade Waltz, Broken Wings, or Valor while rooted as one such example - and both champions are very much like melee marksman, with mobility serving as their tool to deal with range, taking up skill-slot real-estate in the process (where Cait just gets 650 range for free in that regard).
Add a silence against the majority of their kit's power, and the part of their kits that specifically allows them to participate in a team-fight without being so god awful at it they're incapable of doing anything but back-dooring, and you've got even more issues in terms of balance then you solve dependent on how easy it is to apply a mobility silence (which I'm assuming you'd want to be fairly easy, seeing as mobility usually gives an edge in dodging things to begin with).
A Rioter has also pointed out that, while playing immobile champions like Brand tends to feel more like you won because your opponent made a mistake where mobile champions seem to feel more satisfying when you play them well, immobile champions like Brand still tend to bring a lot of power to team-fights and have that reflected in win rates.
Is mobility really out of hand or does it just feel that way?