When I learned systems modelling, we studied spring-mass-damper systems right next to electrical circuits and I have ever since thought of a capacitor as equivalent to potential energy, an inductor as equivalent to inertia and a resistor as equivalent to friction.
KVL states that voltage differences sum to zero around a loop: this is similarly true of velocity differences. This can be difficult to visualize, but imagine a loop of N springs. Assume the loop can have no net translational motion (as above, this means it is not capacitive). Now imagine the different ways each spring can move individually. The loop could rotate as a whole, meaning half the springs have are moving "up" (i.e. have a positive voltage across them) and the other half "down" (i.e. have a negative voltage across them). The loop could be still, but for one element compressing and its neighbor expanding (i.e. all zero voltages with one positive and one negative).
The reason the examples in your systems class worked was likely because the circuits were simple enough that their duals shared the same, or similar, network structure. You can see this with a purely electrical circuit -- analyze a simple LCR loop, and then swap the L & C equations and you should get the same result. This breaks down for larger circuits, where you must find the dual network in addition to swapping Ls and Cs.
There are actually a few different analogies you can make between electrical/mechanical devices. See the top table "Key Concept: Analogous Quantities". This is part of the reason why I don't feel like thinking too hard about what the inventor means when he says the analogy "breaks down" with ungrounded capacitors. http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Analogs/E...
The inventor has created a "new analogy" where the inerter is a capacitor. I read your post as saying the old analogies don't work. Is that correct?
"Ungrounded capacitors" refers to capacitors which do not have at least one terminal connected to the reference voltage (ground). These break the force ~ current analogy because with that analogy, capacitors, which are two-terminal devices, are analogous to flywheels, which are one-terminal devices. The velocity (voltage) of the second terminal is effectively zero.
Your reading of my post is close to correct. The force ~ voltage analogy is incorrect however you look at it because KCL and KVL don't hold. The "old" force ~ current analogy does hold except for the case of ungrounded capacitors, which the replacement of flywheels with inerters remedies.