Thanks @Dave-and-Deb0 for bringing up this important and often vexing issue, and for doing a poll to help.
You are quite right that guests bringing more persons than they booked the reservation for, is a continual and significant problem for hosts. I have seen a huge number of posts on this issue, in the years I have been participating in the Airbnb host groups. It is in fact one of the most common problems that hosts have, who rent out entire apartments/houses. (Not as much a problem for those offering a room in their home, but it still happens even then, with children showing up unexpectedly ). Either the guests "sneak in" extra people, bring children without having asked in advance if it was okay to bring children (some hosts don't accept children and/or have homes that are not "child friendly" and which in fact may have features which are dangerous to small children, such as an unfenced pool or steep stairs) or they bring children and insist that they should not have to pay for these, since hotels don't charge for small children.
There is a considerable misunderstanding in such guests, that booking a host's home, is most definitely NOT the same as staying in a hotel. (I like to keep saying this, because I think it bears repeating, particularly when anti-Airbnb foes shriek that we are "turning homes into hotels": our homes are NOT hotels!)There are many ways that guests may not appreciate the difference between a home and a hotel, but in terms of bringing in extra people or children, we see therein one of the most common of these misunderstandings. I have heard countless times from hosts, in their posts on the forums, that they are frustrated by yet another guest who thinks they shouldn't have to pay for small children, since "hotels dont' charge for that." Guests will even sometimes argue with hosts over this, as though they think that the host's rules are open for argument and can be overcome by argument.
I agree with you @Monica4, that the issue of guests being dishonest and sneaking in extra guests, is sufficiently common that hosts need to have effective methods to contend with this. I think the idea you suggest, of being able to terminate the guests' reservation immediately with no refund, would be an effective deterrent and fair to the host. Airbnb ought to support hosts being able to enforce their own house rules, and one good way to do this vis a vis number of guests, is to require all guests to provide the names and contact information of everyone who will be on the reservation, at the time of booking. Airbnb should clarify to guests that unless host permits them to have visitors over to the house and/or to bring extra guests without asking (some hosts will allow that), then if they bring extra guests when this is prohibited, or bring visitors when this is prohibited, their reservation will be immediately terminated and all fees will be forfeited. Hosts could check one of several boxes to indicate their rules on this issue. Guests would be able to amend the list of those in their party at any time with host approval and/or additional payments as needed. (If host doesn't approve of additional guests they could not bring any extra)
Of course, one of the biggest difficulties for the host who rents out a separate unit listing, is that they aren't right there in the house to see who is there. So sometimes the host has no direct evidence that there were extra guests --- or if they find extra people there, the guest may claim those people are only visiting, not staying overnight, when in fact they are staying overnight. So in order to terminate a guests' reservation while holding back their payment, there needs to be more than a hunch on the part of the host, that there are more guests present than there ought to be.
One may not be able to do much about guests who book for 4 people, check in with 4, and 2 hrs later when the host has long gone, call the other 5 friends to come over, saying, "the coast is clear, the host is gone now so we can violate her rules now and sneak you all in! " And then have 9 people staying in a place that accomodates 4, when they paid for 4.
But at the very least one can do something if a guest has booked for 4 and shows up on the doorstep with a party of 9.