Shotgun is not useful for stealth (in particular close quarters) mostly because of it being loud. You're far better off with a melee weapon and Blitz abuse if you're actually doing a close stealth quarters build in F4. I've done stealth snipers not melee but if I was doing a stealth melee I wouldn't be using a shotgun. And no a shotgun with a suppressor/silencer does not make sense it's a loud close quarters weapon even in FO4.
Actually silencers has another function, it helps stabilize the weapon. Useful with an automatic shotgun. Stealth melee is not always good and in close quarters a silenced shotgun has the following advantages:
1. You will alert less enemies, they have a harder time pin pointing your exact location and they lose interest in you faster. They will also not come after you if they are busy with someone else.
2. Staggering effect in close quarters.
3. It kills more enemies much faster.
So, there is a bunch of ghouls. I will send Preston to stand in the open because he is good for nothing else, the ghouls will swarm him, I can shout ghouls with silence shotgun rather close without alerting and pulling other ghouls to me. If I blitz into the fray, I will be detected even with maxed out stealth.
Silencer in FO4 has close to no disadvantage really. Reduced range... meh, who cares. 
Haven said that, I dont like the shotguns in FO4 so I dont use them. Shotguns should be like those in ME, close quarter 1 shot monsters, the opposite of snipers but it simply isnt.
p/s: In general, going pure melee in FO4 sux. One lousy clumsy attack and that is it - and there is blitz but still it sux overall. Tis better to just take their legs off with guns and then hammer them poor helpless things crawling on the floor. That I like.