Electrician here. Answering in terms of the Canadian Electrical Code, and also my personal safety as an electrician interacting with things that have been worked on by others. I'm going to use the word "Ground" here in the colloquial manner, but technically in electrical, the system itself, where it physically attaches to the earth is grounded. In all other locations within the system, it is considered to be bonded, so you bond the whole system/equipment/building/together, and separate from that, you ground it.
As for the bootleg ground, the simplest answer is that anything at all that causes the system, particularly the safety elements not to function in the intended manner can lead to loss of life. As for whether it is more fundamentally dangerous, some of the time it will be, some of the time it won't. Even with a 120v 15A circuit though, it could mean your life. The most fundamentally dangerous things in electrical are things that you would not expect to either run 50-100mA AC through your heart (which 120V it turns out is just right for for most people) or that can run enough current through you to ensure rapid disassembly. Vastly different amounts of current, but both, ideally, require the element of being something you would not expect.
1.) Single device fault: The intent of the ground wire in textbook terms, is to provide fault currents with a low impedance path to ground.
Inside of buildings, this ensures that should a wire be shorted to ground, the surge of current will be sufficient to immediately blow the breaker (although some brands have batches of breakers that you can literally weld off of while they hum away and fail to pop). On the outside of the buildings, the electrical company is actually depending on every house having an adequate attachment to ground, as this accounts for a portion of the grounding for their entire distribution system.
Inadequate path to ground on a massive (by electrician standards, not distribution standards) distribution voltage breaker can lead to incredibly large failures, dangers and problems.
As for the "grounded case", if it fails to provide an adequately low impedance to ground, it is inadequately designed, and we won't see it commercially in Canada due to it's inability to pass CSA or UL testing. We are still able to order some fairly dangerous objects directly from other countries if we don't know what we're doing, but in order to use them in a commercial or insured product/project, you would need to get an appropriate agency to sign off on them. All of this said, case grounding is well understood and very far from the more difficult aspects of engineering. In countries with better safety codes (Our country looks pretty unfavorably on unnecessary worker deaths and our codes reflect that) every permanent metal object save fasteners etc, and certainly all that would be expected to come into contact with a live conductor will be thoroughly grounded, so long as no one messes up.
2.) Ground wiring fault: To the degree that the situation you are describing can exist, yes, having energized objects that people can touch does present a danger(as a first year apprentice I had to stop an out of country exchange program worker from repeatedly putting his trowel in wet 347V mud. The mud was near a post that had been energized to 347V by an unlucky drilling of a large concrete insert. They had drilled into a lighting conduit for the floor below and either not noticed or they were too embarrassed to report their error to their foreman, so the friendly tile worker showed me by repeatedly shocking himself.)
Freak incidents do occur. If that post had been grounded though, as soon as that lighting circuit had been turned on, the breaker would have popped, removing any danger and revealing that something needed to be troubleshot.
As for the broken ground, the way modern buildings are constructed it is actually surprisingly difficult to achieve in the first place for anything other than small wood frame buildings. In a small wood building it's theoretically possible, but the parts of the connection that would need to be broken are protected by a wall, handled by professionals, and tested before handed off for commercial purposes. This means that while during the construction of a building I as an electrician have to worry about this, as do coworkers from other trades, the end consumer never should, so long as they adequately maintain their house.
The worst I've seen in a building handed over to the customer was probably a broken neutral (also very dangerous) in the building I currently live in, done by who knows who, but likely not another electrician. Second worst was a light switch (not light or ballast) that the owner said buzzed whenever it was on. Someone had not adequately tightened a screw, causing it to constantly arc. They had, however stripped a full inch of wire, bent it around the screw and pinched it with strippers, and that likely prevented something worse from happening.
One other thing is that some safety devices, like GFCIs you find near water or liquids, rely on the ground to function. They're great. They're extremely sensitive, and all they do, is if every bit of current going out on the line doesn't come back on the neutral, they break the circuit. If you get shocked through a functioning gfci outlet or breaker, you will never get locked up, I don't believe enough current can run to stop your heart, and you may not notice the shock.
Lightning arrest systems also rely on grounding systems.
It would be more fair to say that adding grounding to an electrical service certainly averts a vast assortment of hazards, both related to the system and devices connected to it, at the expense of increasing necessary materials, training and engineering costs, and poor installation leading to flaws in the system can still create hazards, in the same way that poor installation could create hazards in a groundless system. If someone finds a way to "acceptably" install something that creates an unwarranted hazard, we upgrade our electrical code. Simple as that.
The fact that it takes some time to think up a scenario in which a grounded system is more dangerous at all than an ungrounded one says something. I could probably come up with a few too for conversation, but I'd rather spend the effort elsewhere and I know whatever I come up with won't make me any less grateful for a grounding system.
If you use any regular ac/dc brick, it is not grounded.
You should not use the iron when it is not grounded.
Why you must ground it?
When you use a "floating" ac/dc brick, there will be charge or coupled AC current (typically half mains voltage) on the DC side of the transformer. Although small, it still has the capability of sparking any other circuitry to death. Just like ESD.
Even if a AC/DC brick has a three prong plug, this does not guarantee grounded DC side.
Explicitly ground the iron or DC side of the power supply yourself.
Many other soldering stations have a large mains transformer instead of a SMPS, this significantly reduces the leakage current to almost zero. You could use these when they are not grounded.
Best Answer
Your equivalent circuit ignores what you'd call Rphase: the series resistance of the phase supply wiring and the internal resistance of the supply itself. The supply cannot provide an infinite current. Rgrounding_wire_resistance will be relatively very small compared to that, so there is a potential divider you haven't shown.
Secondly, a real-world mains socket supply has unavoidable over-current protection in the consumer unit (RCDs/fuses) and (in UK) in the mains plug (fuse). This cuts the supply before something in the loop of the wiring and the short-circuit melts. Earthing is not expected to protect people just on its own.
So Rgrounding_wire_resistance is to be low enough to divert drop as much supply current as possible away from a user-contact area, leaving a low voltage on that area until the protection cuts the supply off.
The Rgrounding_wire_resistance has a minimum limit: what's practical to install (not really thick wires everywhere) and economical for widespread use.