This is not even a contest. A non-GFCI outlet will happily send 1A through hot, into a faulty or misused appliance (think: submerged toaster), from there through you into some other ground (any ground will do, it doesn't have to be that same appliance's ground), without noticing anything's amiss. You, meanwhile, are dead several times over. The ungrounded GFCI-protected outlet, OTOH, trips as soon as the current imbalance reaches 30mA or so. If all the current leaving the GFCI eventually returns, there's none left to be going through your heart.
Yes, the GFCI could fail to open when it should (so test it regularly), but the counter-objection is that having a ground on the outlet doesn't guarantee that the circuit breaker will operate when a loose wire in an appliance touches the chassis. This is why my local electrical code doesn't like the idea of a single failure causing harm, so all socket outlets must be both GFCI-protected and grounded. For some reason I don't quite understand, lighting fixtures are exempt from this, as are stoves/ovens.
It is NOT safe to obtain the grounding from another outlet.
The grounding wire must follow along side the hot and neutral wires of the same circuit for the entire path back to the panel. The reason for this is to minimize the inductance in the event of a short circuit to any grounded metal, either in work boxes, or in the powered equipment itself (such as the multi-outlet box). Surge protection will be much less effective in cases where the surge is a differential between hot/neutral and ground (which actually is a common type of surge) due to this same inductance.
Inductance is minimized when the wires with the current going in both directions are close together. Their magnetic fields overlap and mostly cancel each other out (there will be a field inside the cable between the wires). When the wires are far apart, their magnetic fields will extend a greater distance and there will be more inductance. Also, metal located inside the "loop" between the wires can be affected by the magnetics (sometimes in a hazardous way).
To correctly upgrade a 2-wire circuit to 2-wire-plus-ground circuit, the cable must be replaced with one that has integrated ground. In some rare cases, the proper cable may have been run, already, but an ungrounded outlet was used.
Simply running a single wire along side the old cable is also NOT safe. The wires of a circuit, when conducting current, will try to physically move apart from each other due to the orientation of the magnetic field. Physically binding the new single wire to the cable with cable ties not further than 6 inches apart for the entire length could avoid that issue (but is still not code compliant). If you are doing that much work, just replace the cable with the proper type. FYI, this was one of the hazards of older knob and tube wiring. Single wires inside conduit are known to move around, but the conduit sufficiently confines the movement so all you get is some noise.
In many cases you can obtain much (but not all) of the safety of grounding with the use of a GFCI outlet. The grounding wire still cannot be used in this case. It just gives a safer 2-wire load. It is not sufficiently safe for appliances that have frequent human contact, like a computer.
Best Answer
(NB: I'm not an electrician, just a homeowner who has done some electrical work)
Related: Does a non grounded GFCI meet code?
It is 'perfectly acceptable' in that it is compliant with code (as long as it is properly labeled, as you state). A GFCI if wired correctly should trip if there is a difference between the current coming in and the current going out (between the hot and the neutral, basically).
However, you are right as well that there are dangers/disadvantages to not having a ground wire. The point of ground wires (and requiring them in new construction) is because it is safer both for you and for the equipment you have plugged in.
Without a ground, a lightning strike or other power surge can seriously damage you (if you are touching a plug or outlet when it occurs) or your equipment plugged into any ungrounded receptacle. Ground wires act as an 'escape route' for extra current to travel safely back to the actual ground underneath your home.
Millions of people lived for decades in non-grounded homes and never had issues. But many people have experienced shock or equipment failure due to said lack of ground. It's up to you if the threat of power surge / lightning strike ruining your equipment is an 'acceptable risk' or not.