Neutral in the context the OP describes means they neither consider you friend nor foe and has nothing to do with Law/Chaos or Good/Evil.
That was me being funny. Sorry for confusing you.
If you are a Guard in a throne room I would consider you on guard for dangers at all times unless otherwise noted.
If you are the DM, great. That's your call.
Personally, I don't think being alert 100% of the time is humanly possible. Hence, a possibility of being caught off-guard if someone doesn't look like a threat. But perhaps my perspective is influenced by my ADD. It's certainly not possible for ME to focus on one thing for extended periods of time unless it's one of my personal interests. (I had a job as a security guard once. I learned that I make a terrible security guard and any would be thieves could load up a moving van with your stuff without me even realizing it.)
They don't suspect you of being hostile but, since they are neutral towards you, they consider you possibly hostile.
See, that's another bit I disagree with, and what I was referring to when I said you were assigning a meaning to "neutral" that I wouldn't consider.
You seem to be saying that "neutral" means "on high alert". I do not. Neutral means they are in their default state of alertness. That is determined by the DM, based on what they know about the guards and their assessment of the situation. There is not a universally true answer to the question "can they be surprised", other than "it depends".
Given that they are escorting a motley crew of armed individuals, the DM may decide that they are on alert and can not be surprised. (This seems to be your default answer, but it's not the only one.)
The DM may also decide that this is a peaceful, progressive kingdom that considers adventurers a valuable part of the economy, and armed groups of individual are wandering around all over the place, and the guards don't expect this group to be any trouble. Suprise is automatically successfully, because no reasonable person would attack the kings guard in broad daylight like that. The ensuing capture and execution may also be automatically successful, depending on how much of a dick the barbarian's player is being.
Or the guards may actually be "neutral" and neither be overly alert or overly lax, and the DM will call for a Deception vs Insight opposed check.
I'd consider the possibility of being surprised by being attacked by the barbarian (surprised in a 5e mechanical context) to be zero.
You are absolutely entitled to your opinion.