I finally got past this after about 30 attempts. I'm playing on Xbox 360.
What worked for me was to rewind. I didn't kill Danny, and I identified myself as Lee by asking about Katjaa and Duck. I don't think these choices affect anything, but just in case you want to try everything I did, there it is. I don't think Tell-Tale-Games would mess something up like that.
Basically the glitch for me was that Mark at the top of the stairs didn't load. I'm fairly certain he's visible as soon as you open the screen door. If you don't see him, you're going to get shot. It's not red light/green light where you move when Brenda is looking at Katjaa That's a load of crap, basically. It's either bugged or it's not, and Mark not being present is a prime indication.
I did end up switching controllers. I'm not sure that the problem is the controller not resetting to null, but it couldn't hurt to try. When I clicked it forward, what I did was tap the stick up, but not the whole way, basically just enough to get Lee to edge forward - basically halfway to the top. Trying to quickly flicker the control stick the entire way just got me killed.
I think the problem is improperly stored variables, not the null stick position. Basically someone programmed too late into the night, and their testers just happened to get through it by luck of the draw so it slipped through.
I severely doubt that this is actually a test of skill - to have a situation this sensitive is far, far beyond the spectrum of the game's difficulty, and out of the scope of the rest of the first two chapters. This isn't a skill game, and shouldn't be one. Honestly, I would prefer if it was a skill/timing based game, but mostly it's just guiding an awkward cursor onto light up beacons. If this is just programmed to be 'difficult,' it was a major oversight, and should be patched down to the rest of the game.
Think about it: Why would they have a difficult stop/go game for the penultimate encounter with zero instructions, and then for the final scene they give you specific instructions on how to punch. There's no way that it's rigged to be this difficult.
They may be delaying patching this simply because they are in a rush to get future episodes out. This bugs seems sort of common, but if it happens to 1/1000 gamers, financially it's not worth spending time on to build a patch for.
Personally, I'm not going to buy the next episode for at least a few weeks to make sure that there isn't a problem like this. I love Walking Dead, but not enough to sit through this kind of frustration again.