Sadly, I'm not sure we will ever know the truth. From videos I saw, it appears that protesters did attack police and overwhelmed them. Meanwhile, they cleared the capitol of representatives to keep them safe. At some later point, the police stood down and allowed people to go in.

I do not believe the police extended an open invitation, but rather they just didn't try to stop them.

The problem with charging people with trespassing is that I am quite certain that some of the folks who were late to the party (i.e., they were never there confronting police) just saw open doors and people wandering through the doors. How would they know they're not allowed in? I am sure some of them really had no idea. If I saw open doors with police standing to the side and smiling as people walked past, even I might assume it was OK to enter.

But as I said, we'll likely never know the truth.