I finally figured this out, I think.
The reason the WMI repair thingie works is simply because it turns off the "Forceware Intelligent Application Manager" service, which is nSvcAppFlt.exe. This is nVidia's built-in firewall, which is installed on any computer using the nVidia chipset, which is a hell of a lot of them.
This is on top of the Windows Firewall. The hidden nVidia firewall defaults to "on" even if Windows Firewall is turned off. IMHO, having multiple firewalls running is stupid and a waste of resources, especially in a corporate LAN environment, but even if you disagree with that, you definitely need to turn it off or reconfigure it in this case.
You can turn it off by going to Start | Programs | NVIDIA Corporation | Network Access Manager | ActiveArmor Firewall. I don't know why this thing has so many different names.
I'll bet if you check, the computers that are having this problem on your network (Lenovos, for instance) have nVIDIA chips, while ones that don't don't. Or perhaps they have some other motherboard chipset firewall installed; I've only found it on computers with motherboards with nVidia chipsets -- in our case ASUS motherboards with AMD processors, but there are others.
Motherboards with nVidia chipsets used to give you the option of installing or not installing Forceware when you load the drivers at initial setup, but it looks like they don't anymore -- they just go ahead and install it.
There's probably a way to configure it to allow Lansweeper to access RPC, WMI, etc. without turning it completely off, but I don't know what it is, nor do I care.