It's not a privacy concern, I just would be leery about recommending corrective work on a dog unless I know how far he's going to take it. If he's all noise and easily cowed, I'd just make it a two person job for now, sit him in a corner where he can't back up, open his mouth by grabbing his upper and lower jaw and curl his own lips over the teeth and hold them open. If he's truly fearful, he's more dangerous, and I'd worry about pushing him. Your faces are very close to him then and if he snaps, he may go for your face. But if you have a jaw in each hand and hold it open, he can't overpower you.
If he's just lolling and squirming, then have the other person, with gloves on if you're worried about it, stick it right at the back of the tongue and throat and push down. Have them pull the hand back fast after pushing down then let go, because it'll force a swallow and impulse to close his jaw.
He's never gonna love it, and the more you punish, the more resistant he'll get. Over all, if this is a daily med, I'd find something more pleasant for both of you, give him a treat, pill, and treat to finish so you don't leave him with the yicky pill taste.
There is a more corrective measure where you actually pull his own tongue out so he has to bite his tongue if he bites, but that's more of a fight than I'd have you do. I'd just try to desensitize him by working on getting in his mouth a couple times a day with no pill.
He seems to tolerate the joint supplement, which is the long term one.
He didn't pull away from me this morning, but growled and started to snap. I don't think he's afraid. Just stubborn.