For splitting kindling, and up to small logs, the traditional tool is the froe.
That might be what you meant by a "cleaver"(?) although many other kinds of tools could be used with fair effect.
The handle gives good control, and by twisting with the handle small strips can be easily worked loose. The tool also lets one make "lath strips" and/or shingles - or other thin pieces with relatively good control over the thickness.
A good tool for the woodworking shop, as well as for the firewood lot, and a fairly heavy one might be adequate for splitting down half- or quarter-split logs for a smaller fireplace.