I also noticed this problem. The first thing I did was right click on system.log and selected "reveal in finder". I then looked at the properties via "get info" and noticed that admin had read-only access. I changed this to read/write and that eliminated the problem of viewing the system.log. It also made me suspect that this is related to not being logged into a user account with admin rights. Changing the system.log file isn't a solution because the macOS switches this file out every day. I am sure the new one tomorrow will have the old access privileges.
I then logged into my admin account. Opened the console app and clicked on Macbook Pro and there was the old running list of console messages just as pre-Sierra.
Not sure whose hair-brained idea it was to make the console app only work on an admin account. It makes it much more difficult to diagnose problems. I only use my admin user account for very special reasons. Switching to the admin user to use the console app when I suspect there is a problem does not make sense.
I know it has been some time since you created this question, but I would be interested to hear what you have found or if you tried the console app in an admin user account.