Well, I have worked on reservations, ticket desk, check-in and as a gate agent prior to becoming cabin crew, so I have a (basic) understanding of the processes involved.
Dispatcher - your argument seems a little circular to me - If the handbaggage problem doesn't exist, then that is due to the ground staff exercising good judgement. If there is a handbaggage problem but the cabin crew resolve it, then that too is down to the ground staff exercising good judgement. If there is a hand baggage problem and the cabin crew can't resolve it then they are damned for leaving handbaggage on the jetway and adding to ground staff workload! I am unclear as to how cabin crew are supposed to "win" in this situation.
Cabin crew can't tell for sure that the cabin will run out of space until it happens. In other words, you don't discover that bags will have to be left on the jetway until the very last minute, when the space runs out. Up until then, we have been attempting to find hatrack space and/or persuading the customer to part with their bags ("they let me have it last time..." etc. etc.). On large aircraft, the last opportunity for hold stowing bags is not up to cabin doors closed, but in fact well before - when the last hi-loader is pulled away and the holds are closed. Our customers know this as well as we do, and this makes them even less willing to part with their baggage.
Noting your comments about short staff and "headless chickens" I sympathise, but would respectfully pont out that cabin crew tend to be in the same position these days. Workload during the last few minutes before doors close can be high for us too. Captains tend to become stressy when you tell them that departure may be delayed due to excessive hand baggage (not surprisingly) - remember that you don't tell them this until the last moment because that is when the problem becomes visible.
Cabin crew will tend to moan at the dispatcher only because he/she is the ground handling company representative that is there at the time. If the gate agents were to wander down the jetway and ask for feedback, I would be delighted to provide it to them directly, rather than via the dispatcher. They don't do that, because they are already on their way to the next gate or back to the office.
The fact remains that the gate agents are responsible for controlling the volume of handbaggage permitted on the aircraft. They can properly see what the passenger is carrying (cabin crew can't except at the immediate entry doors because of the seats) and can remove it in good time for it to be hold stowed. The airline contracts and pays for ground staff to control passenger baggage and hold stow those items that require it. A failure to do so is a (technical) contract breach. Excess handbaggage is often directly related to excess baggage payments on the checked bags, and therefore a failure to monitor and hold stow, and to apply the appropriate excess baggage charges is also a denial of revenue to the airline involved.
Certain airfields are better or worse than others - most cabin crew could list destinations where they know they will have a problem. This is usually due to cultural factors, laziness or lack of relevant skills by the ground staff. It's not just third world countries either - try Italy on a busy day, or Americans with suit bags for proof of this.
IMHO, the best solution is cross-training: If cabin crew were to work the gates occasionally they might have a better understanding of the problems involved - vice versa the gate agents. Everyone is an expert on everybody else's job, until they have to do it! This is known as the "People Express" solution, and it develops workers as teams rather than as teams in conflict with each other. This kind of training could take place during quieter, shoulder flying seasons if the will existed: Sadly, it doesn't. That's why I brew a pot of tea when I can for the baggage loaders and pushback crew (the few times when I can) - because I've done those jobs too for a short time, and know how very cold, wet and miserable they can be, and how any warmth or concern is welcome.