Just because something belongs to the people does not mean that 1,000 of them need to be able to rush into it.

Dutch Parliament has a visitors entrance and is (in non-covid times) easily accessible to the public. But for obvious security reasons their private offices are behind these kinds of locked doors. Since a few years I think you have to go through a metal detector to be allowed into the public areas.

It makes no sense at all that the US Capitol doesn't have stronger barriers between the public areas and the private offices. Every bank or other large company has such a setup for information security reasons.

The offices are public areas because they serve public officials carrying out public office.

But why does a public official's office need to be a public area? I'm sure you're not allowed to walk into it without an appointment, so having appropriate security barriers wouldn't stop them doing their job at all and would improve security.

Without referencing any source I'll just assume you made that up. As far as I know there's no difference between say the French or German republic or the Dutch (symbolic) monarchy in this regard.

Dutch representatives are accessible by the people. They have a walk-in hour, you can call them, email them, write them, you can join hearings and meetings where they're present, they go out into the country to talk to citizens. But what you can't do is waltz into their office. This has obvious reasons in a post 9/11 world, and it has nothing to do with the fact the Netherlands has a king who has a purely symbolic function and does not participate in politics, no different from say France which is a republic, or Germany which saw a mob storm the Reichstag a few months ago and was easily held off by the police, which is also a republic.