With no weight so to speak I wonder how it'd do trying to push down to get a bucketful of hard clay. Be a fun toy if you've got the bucks!
This is a common misconception...
The short answer is, they will do fine because they have outriggers and do not need to rely on weight. As the unit is digging, it forces the outriggers harder into the ground. My log trailer does the same thing with its outriggers, and the pictures showing the mini-excavator by the original poster depicts a blade on the front. As long as the operator was doing the digging with the blade in front, it will have plenty of digging ability.
This is far different then my stand alone excavator (Hitachi 160) because it has no blade or outriggers. In that case it uses its counterweight on the back to counteract weight transfer while digging, but it is a balancing act.
Most equipment has this. For instance, a wheel loader has 60% of the weight on the rear tires, and 40% on the front tires, until it gets a bucketful of dirt, when that happens, 60% is transferred to the front, and 40% is in the back. This weight transfer is what sizes the capacity of the bucket. It is why I cannot put a 10 cubic yard bucket on the John Deere 744G; it would be unsafe to operate.
The same weight transfer takes place on a skidder in terms of operating empty, and with a full twitch of wood behind it. It would not pull well if the front tires were off the ground.
All this is tractor science, with drawbar pull, tractive effort, and weight transfer all engineered out so that the machine is efficient at what it does.