Yes, on both counts.
His problem as a Liverpool player in the long run might be that where he is strongest (an attacking midfield player or on the wings) is where we are already strongest as a team. Klopp surely will never prefer Henderson to Coutinho or Lallana, let alone Salah or Mane. I'm pretty sure he wouldn't be ahead of Oxlade-Chamberlain or Milner either - and before you can say Jack Robinson we'll have Woodburn in the mix too.
Henderson's skill-set, oddly, might be liberated by playing for a team that sits back and absorbs pressure and hits sides on the break. A team like Newcastle perhaps? Or Huddersfield. He adores attacking empty space (a rare commodity when you play for Liverpool). And he prefers the decoy run to actually having the ball.
Henderson should really get a chance in the Wijnaldum role, with someone like Can doing the job at 6.
It would suit his skillset better. He's a brilliant presser of the ball and you do much more of that as an 8 rather than a 6.
He can then use that energy for those counter attacks as you mention. But where I disagree with you is that is the only time he can contribute. The season before Klopp came he was our most productive player in terms of goals & assists combined in the league - yes more than Sterling, Coutinho, Sturridge etc. He didn't just get them due to goals on the counter attack. He's got a superb cross, he did some lovely flicks in tight situations around the opposition box.
I genuinely think people forget how good Henderson was that season and how much better he is in that box to box role.
He's 27, he says he's over his injuries, he seems to be able to play as more of a box to box player for England. I still think he can do it physically, I just feel Klopp really likes him in that deeper role - as he's been the clear 1st choice for that position since Klopp has arrived at the club. To be fair, until Henderson got injured last season and we were near the top of the league with the balanced CM including Lallana, not many had too many issues with Henderson - despite it probably not being his best position. Last weekend for example he had a really good game too.
He gets a hard time and although he's no Alonso or Weigl - it doesn't mean he's poor either. There is a middle ground and I personally think he gets too much of a hard time when we don't win as he's always the first to be blamed. Often when he's one of the least at fault.
Mid week the midfield were collectively awful. I feel sorry for the deepest CM, as they are really reacting to what the two in front do. The two in front should be doing the bulk of the pressing, forcing errors and getting the tackles. The deepest one should be more of a interceptor, plugging the gaps and spaces. If the two in front of Henderson don't work, then he probably won't either as they need to be in sync collectively. They clearly weren't, but they were on the weekend.