If I understand your complaint correctly, Painterroy ... you've got a couple of points confused.
I'm guessing that this Help Screen is the one to which you refer.
At the bottom of this Help Screen, members of groups are exhorted not to use the message system to invite, but to use instead the special Invite People button. This admonition does not imply WHO is empowered to invite, just the method that should be used. Many groups, like the one I am super admin for (Blackmar-Diemer Gambit), have OPEN admissions - anyone can join. Other groups DELIBERATELY want to restrict memberships to by invitation only.
I'm guessing that what you are asking for - using the current implementation - would be groups which a) limited membership to invitation, BUT b) gave all current members the right to invite - essentially making all members admins. In my view, this is implementable by simply asking the admins to make all incoming members admins (although I'm not sure if chess.com imposes a limit on this). Failing this, the group could be changed to anyone can join, since you're proposing nearly that by allowing ANY member to invited anyone else.
I'm sure all I've done is confuse the issue. Oh well.
In the message rules, it states that you are not allowed to ask someone to join your club. That you have to use the invite member on your admin section. That seems unfair if you are a member of a club and want someone else to join and your not the administrator of your club.