Update: John Ward talks about this exact situation in a video. It struck me how similar John Ward's death diagram is to your diagram above, complete with GFCI (RCD) in that exact spot. It applies to "whole house GFCIs" in British houses with no ground rods, but it's the same concept exactly.
A relatively recent branch exists where the installer used modern NM-B 14/2 wiring for all outlets in the branch circuits, connected all the grounds to the outlets, but left it unconnected at the junction from old-ungrounded to modern 14/2.
NEC 250.110 prohibits extending a non-grounded circuit.
However you should check if the old work is metal conduit or metal jacketed cable, which is often a valid grounding method, which means the old boxes may be grounded after all.
Also most places it is illegal to do electrical work without an electrician's license, which is a 5 year apprenticeship, however it is OK if you are the owner-occupant of a single family home.
Also most places, a circuit extension requires pulling of a permit.
Sounds like the last guy left you a real mess.
An "island of grounds" is dangerous but GFCI helps
Because, when you have outlets with ground attached to each other but not to the panel, it does not arrest ESD or ground faults. It shares them with everything in the island.
So the laser printer throws a big spike. The power strip dumps that spike to ground. Except all that did was make sure all the grounded appliances received the full spike!
A PC power supply develops a bolted fault to ground. OK, now everything in the island that is supposed to be grounded is energized at 120V.
However, adding a GFCI in the first position at least renders it far less unsafe.
If you bootleg ground to neutral, it isn't ground anymore. It's neutral.
You say "but neutral and ground go to the same place!" Sure, when everything is working properly.
You may notice that neutral is insulated. Ever wonder why? Because there are very common conditions under which neutral will be energized.
So let's look at what your change really is. I've drawn the GFCI protected zone in green. Note how one wire color has been changed to reflect what it actually connects to, and here you can see the reckless and foolish thinking of bootlegging ground.
Note that while the hot and neutral are inside the GFCI protected zone, the, um second neutral is not inside the protected zone. So a neutral problem in the "rest of the house" department (or even off diagram) could energize neutral and light up all the "things that are supposed to be grounded" on the equipment. And what would the GFCI do? Even if it tripped, it does not disconnect the second neutral.
Neutral and ground must bond at exactly one place: the main disconnect, where Code says it must.
It's dangerous to take "a little bit of knowledge" and try to extrapolate that into a prioritization of safety risks. The fact is, humans are infamously terrible at that. That's why we have a science and data driven approach called the Electrical Code.