Please don't automatically assume that if the crew say no to your own restraint or seat it's necessarily because they are lazy or just don't want the hassle. Crew do NOT make the procedures and policies they are required to uphold.
My airline only allows passengers one type of 3rd party child restraint and we are NOT permitted to fit it for you. As to why some airlines will allow it and others wont has zero to do with how nicely you chat to the ground crew. And has more to do with things like this...
The level of risk the airline is willing to assume. Unfortunately there are people who will take no personal responsibility for anything and will use any incident as an excuse to sue an airline. Which may include injuries that result from them incorrectly fitting their own child restraint or using one that is not designed for the purpose they have used it.
Space in the cabin. Lets say you have 200 passengers boarding a narrow body (single aisle) jet on a tight time schedule and 3 passengers hold up boarding while they stand in the aisle for several minutes fitting a child restraint. On some jets it's just not possible to fit a child restraint quickly and without standing in the aisle. Preventing everyone else from moving past you. Many airlines have tight turnaround times that really only permit pax walking off, a quick clean and pax walking on. And what if a parent bought a car seat onboard then decided it was better not to use it. Where would it be stowed? It may not fit in the overhead lockers.
Additional training required for crew. Crew are responsible for ensuring you follow safety instructions on board and know how to use the equipment fitted for your safety. As a passenger you will rarely be aware of the amount of equipment we have, and we have to know how to use all of it. Some airlines may find it unnecessary to also train their crew in the correct use and fitment of various child restraints. How many choices are there on the market? Should be all know how they should be used correctly and be able to check you have fitted it right for the protection on your child? As crew I am always very concerned for little ones on my flights and that their seatbelts will serve them in an emergency.
A car seat is not an aircraft seat and a car child restraint may not work as effectively on an airline seat as it would in a car. Aircraft hit the tarmac at 300kph or faster. The seats on an aircraft are designed to take stresses differently to those in a car owing to the greater forces and different centre of gravity. Your car child restraint may be rated to withstand the forces in a car but not in an aircraft causing it to break away in a crash. Your car child restraint may attach to the weakest parts of an aircraft seat, or inadequately (especially if it's the type that requires an over shoulder sash belt to be secured) also compromising it's use. The best survival position for your child may not be fully upright strapped to a seat. And finally if an aircraft is on fire you have less than 90 secs to get out and survive. That's why seatbelts are so easy to remove, even in a fully dark cabin filled with smoke and rife with panic you have a good chance of undoing your seatbelt quickly and getting out.
Passengers have little situational awareness on a flight. I witness this daily as we are just minutes from touchdown and someone inevitably gets up to go to the toilet. They are not trained to listen to PAs, aircraft noises and chimes the way we are. Likewise they are not trained to evacuate the aircraft with a personal child restraint either and may panic, need assistance or end up entangled in the thing. Conditions which crew won't have time or opportunity to deal with.
I know that we all want to think about comfort and convenience when we get on a flight. Your crew want you to be focused on that too. But crew have to be focused on the one in a million chance that something may not go to plan. What you bring onboard doesn't just affect you, it also affects everyone around you. Any item that creates an obstruction to exiting the aircraft quickly is a serious danger in a confined space with lots of people.