This gist of the actual answer is what Cheffo Jeffo said. Older monitors simply take the input AC line and dump it into a rectifier to generate many voltages on the board. This means that both power ground on the monitor and B+ (as well as other secondary voltages) are all referenced to the incoming AC line. This means that they are all (including the metal frame of the monitor) "hot" in that they are live relative to earth ground, and one can pull a lot of current off of this. This will happen no matter which way you hook up the AC line due to the way a bridge rectifier works.
Clearly, this is problematic. It's a safety hazard, of course, and if your cabinet has power ground tied to earth ground (like many PCs do), you'll basically short out your AC input via your video wiring, monitor, and arcade boards. That's likely to result in a pretty bad smell and some smoke.
The isolation transformer allows the AC input to "float" relative to earth ground. This makes it safe to tie the frame of the monitor to earth ground, making everything safe to the touch. It also removes the possiblity of the aforementioned ground loop, removing the smoke problem.
TVs often just deal with having a "hot chassis", and they use various techniques like high frequency AC coupling on video inputs (especially on TVs with only RF inputs) and plastic enclosures to make it safe. An isolation transformer is standard equipment for most TV repair techs because it makes it safe to work on the monitor while plugged in, even though normally touching just about anything inside it would zap the bejeezus out of you if you were in contact with earth ground.
Monitors with a SMPS (switch mode power supply) can easily incorporate the necessary isolation into this power supply, and they therefore do so.
The inrush current and HF noise injection problems are just minor bonuses of using an isolation transformer, not the primary purpose.