This is sort of my quandry. Is there really any point in starting with a small cleaver if in a few months' time I am going to abandon it because it is not really a proper cleaver? Will small cleaver techniques even help me at all with using a large cleaver, or does it just allow me to clumsily adapt gyuto techniques and delay learning real cleaver techniques?
A small cleaver is a proper cleaver. Using one, a person will get a good understanding of what a cleaver can do.
The nakiri met most of my needs, until I had to prep large amounts of veggies for family gatherings. I went to a small cleaver, but it didn't make much of a difference. I tried a gyuto, but didn't like, that the flat area was less then my nakiri. Plus having a tip, wagging around, in a kitchen with kids running in and out. My hunch was that a cleaver could be the answer to my needs, small kitchen, large prep. Did a search on Google, and found the old forum. Talked to the cleaver users, and decided to try a large cleaver.
There is something about large cleavers. At first I thought it was the size and weight people objected to, but its more then that. A person using a gyuto for the first time will find it to be a liberating experience, its light, thin, sharp, and easily makes most cuts. A cleaver is not light, but it is still thin and sharp. Instead of being liberating, the cleaver demands to be used in a certain way.
The cleaver is the best knife, for chopping and push cutting. That being said, the gyuto is not that far behind.
The biggest advantage to using a cleaver is how efficiently it cuts. The weight means the knife isn't going to move unintentionally. Lift the knife, put it over the food, and the weight will do most of the cutting. The size of the knife, means I can put my knuckles against the side of blade and cut as fast as I want, not having to worry about drawing blood. I feel that I can use a large cleaver, over a longer period of time, then a gyuto, because it is a much more efficient cutter.
In your post, you mention that most of your knife work is vegetable prep, and that the majority of it is fine dicing and mincing. The knife that worked the best for fine dicing and mincing was a nakiri. The cleaver comes close to what a nakiri can do.
You also mention that your chef's knife doesn't handle well large quantities of onions and greens. If your chef knife is a gyuto, it should easily be able to deal with onions and greens. If your chef knife, is from a German company, you really should look into a gyuto. Its a much easier knife to use, and does 95 percent of the cuts, needed in the kitchen.
Large root vegetables and squashes are easier to cut with a thin gyuto or sujihiki, then a cleaver. There are some squashes, that are hard enough, that they get pounded open with a meat cleaver.
Don't get me wrong. For chopping veggies or prepping large amounts of veggies, the cleaver is my go to knife. Hopefully I am giving you an idea of what to expect from a large cleaver.
Jay