You can have both: they take up two different Attachment slots. Silencers are Muzzle Attachments, while vertical foregrips are Bottom Attachments.
That makes more sense. In the video, I thought it was labeled as, "front attachment".
A silenced shotgun is the equivilent usefulness to a saw off sniper rifle
Firing shot out of a silenced shotgun is a terrible idea. Firing sabot/slugs out of a silenced shotgun = much less terrible idea.