Is this a problem
I think Rory Alsop's answer is generally correct. The more highly-active users we get, the healthier the site should be, both through diversity of questions and answers and diversity of moderation. It's almost certainly worse for a new user experiencing close votes / deletion to always see the same names coming up on those votes (if they can see them); I know it's discouraging to be one of those people voting "close" or "delete". I don't want to feel like a strict teacher or a scolding parent. I just want the site's standards to be upheld. I know I was lurking here and finding it useful before I started posting in 2015.
Kim Holder's answer on "Be welcoming - a new Space Exploration Stack Exchange" is, I think, a good perspective from an experienced user on how that barrier forms between the "in group" of active users already on the site and new users, and some ways the "in group" can help keep that barrier permeable.
Like any community, though, for the new user that's going to boil down to not only interacting frequently with the community, but doing so in a way that community accepts. Frequent interaction in ways the community doesn't accept leads the community to drive that person away. If we don't try to keep StackExchange's format and standards, we may as well all just go to /r/space or /r/spacex (which will also have their "in groups").
However, while more highly-active users would be better, I'm personally still impressed by the breadth of questions the community answers and the depth with which we're able to answer them. So I also don't think the site is suffering or at risk from the small size of the in-group.
What are the main problems this creates
I think there can be some distortion of The Real Rules into The Perceived Rules. There are comments floating around to that effect all over on the site. I dare say most of us try not to have that happen--see a number of Meta questions on if the community agrees with some moderation action or other, and the relative infrequency of mods having to come in and undo community moderation.
Referring again to Kim Holder's answer
weariness has set in for [the Blender SE community] when handling their high number of unclear, very confused, or unresearched questions, opinion questions, spam, jokes...
... it's worth noting, the grind gets worse as sites grow.
Space SE is still tame relative to that site. It's only a couple times a month that I see a post that's just some jerk having a laugh and it needs to get deleted immediately. The thing that wears on me more are folks who are just always going to have to be policed; their questions won't make sense, their answers either won't make sense or will be incorrect, and then they'll get in a fight with you about it when you point it out or try to fix it. It seems like there's always somebody like that, and that's probably a bad thing. I try to moderate less when I get that "us against the world" feeling, but I bet it happens to more people than just me, and surely that leads to unnecessary conflict with a new user who just didn't read the rules (I didn't either! I just hopped in and tried to contribute in the same way other people seemed to be, and that mostly worked).
How should this be dealt with
Again I concur with Rory. It'd be great to have more people be highly active contributors. I think we need to convert new users to active contributors, and get back active contributors who have left.
Unfortunately, a lot of this is basically volunteer work. It can be rewarding, but also annoying. I don't know how to get more people to do it, or retain them. I tend to float in and out depending on how busy I am with other things.