This issue is very complicated, but I will do my best to explain it.
Originally I discovered this bug without having the ability to post in #bug-reports and instead made this topic. Please note that our results of debugging were rather inconclusive.
After attempting these client-side ragdolls that are connected (via Weld) to a Server created RootPart, I have made this observation.
The rigs that are made using a MeshPart (like in the original post) have the unexpected and strange network-ownership behavior. RootParts are out of sync between Clients, Server, etc.
Attempting the same methods, but using a rig that is assembled with Parts (as opposed to MeshParts) you get these results. RootParts are in-sync with clients and the server but physics updates do not happen on the Limbs that are made on the client.
Swapped out the usage of a weld with usage of an AlignPosition and AlignOrientation and the results are still the same.
Expected behavior
The expected behavior is for the ragdoll to behave like a proper ragdoll on the client, with the instances (limbs) created on the client behaving like proper physical bodies.
The Solution
For those of you who want to achieve client side ragdolls still, here is the best way I found. When the character is ragdolled, have an Attachment in workspace.Terrain and set AlignOrientation and AlignPosition Attachment1 to that attachment. On Heartbeat, set that Attachments CFrame to RootAttachment CFrame. This should however not be considered to be a permanent solution, as it does not move as fast as Humanoid physics resulting in internal lag.