So, Sinjin just murders Kent whenever I start to close in and I think I know why. I cleared that Hospital before but left all the Protectrons "alive" and functional. Something tells me that the glitch must be one of those Medic Protectrons in the basemant that trigger Sinjin and co to open fire.

I can kill Sinjin in one shot with my crit that I've saved up, but I cannot get there in time even if I just charge through the Raiders.

Would turning the difficulty down help?

Also, I might reload a save from an hour or so before (when I had to kill Smiling Kate and that other guy) and go to the Hospital early and kill all the Protectrons. I mean, they're the only two options I can think of at least. :/

It can be done, but it's tricky. I tried several times. IIRC, there's a dialogue option where you call out Sinjin. None of my efforts to kill him first (without speaking) were successful.

There seems to be a fairly common glitch with that part of the quest. I've never experienced it, but I've heard almost the same thing mentioned by several other people.

You might want to check out the Wiki page for the http://fallout.wikia.com/wiki/The_Silver_Shroud quest. Specifically #21 in the objectives, and then the stuff in the bugs and notes sections.

That said, that is a really difficult moment in the quest. With my first character I had to reload it a few times before I was able to save him. With my second character I knew how it was going to go. I had Better Criticals and Critical Banker both maxed out though, so I just popped some Jet right as the doors opened, immediately opened VATS, and banged off 4 criticals in a row. It took care of things quickly.

Those robots are all hostiles.
Best option is to put them down.

One can provoke Sinjin to attack you first instead of Kent in a dialogue option as well.

I was able to take him out from the elevator with a couple sniper shots, but his sidekick then shoots Kent. I've given up on saving Kent to be honest. Would love to save him but I think the quest is just broken. It really shouldn't be this hard.

I just responded as the Silver Shroud every time. It went fairly easy. Of course I was loving responding as the Shroud the whole way through that one. One of my favorites, for sure.

edit: in the wiki that would be #22, 3. All of the henchmen fled, and I took out Sinjin in VATS with a headshot critical hit. The henchmen remain in the room, but cower and do not engage in the firefight at all. Also, to the OP, yes turning down the difficulty will help. It will make it a little easier to kill Sinjin with the headshot critical hit. Pop some Psycho before entering the room to up your damage dealt and you should not have a problem getting through it with Kent still alive. Also, be sure to be dressed as the Shroud, I haven't tried this method without being dressed as the shroud. I believe it is required.

This happened to my girlfriend, it's the robots. I think all she did was try to be undetected, and save often, like way often. Took a few reloads, but she was able to make her way to the basemant and finish the quest without reverting to an earlier save

Yeah, I'm being stealthy too. I've got just past the second elevator which takes you near to the entrance from on top of the nearby parking lot outside, but Sinjin opens fire every time. I'll let you know if I work out a method to do it.