I'm rather more concerned with how it was supposed to work at all... at first I thought they were wired as complementary source follower, which sucks because of the high drive voltage required (and V2 isn't labeled so who the heck knows what's going into it here), but come to think of it the P and N are on top and bottom so depending on which way the P is oriented (the symbol is conveniently missing the substrate connection so it's impossible to tell D/S apart), it's either a PMOS working into a diode, or two whole-ass diodes in series shorting out the supply.
To be clear: the simulator doesn't make any assumptions, or efforts towards realism (besides certain gimmicks in rare cases), the model is entirely up to you. An ideal voltage source will not click off when you short it, nor will components emit magic smoke when you overpower them. You must check yourself, whether currents are realistic, power dissipation is within limits, etc.
What you should have, is a CMOS inverter: N source to GND, P source to +V, gates tied, drains tied. With the transistors and supply voltage shown, this won't be too bad, but will need somewhat higher drive voltage (say +8/-3V) to perform properly.
The caution about shoot-through occurs when the gate voltages are intermediate. For these transistors, it won't be much, because they are barely on at 2-3V, but, doing this with representative types that you'd use in new design, you'll most likely have logic-level types that are barely on at merely 1V, and strongly conducting by 2-3V. So over the 1-4V range of gate voltage, you have a problem. As the gate voltage swings through that range, in the whatever 10s of nanoseconds it takes, huge supply currents will be drawn (maybe 10A+, depends on type and etc.).
And yeah the capacitors are just wrong. You probably get a simulation result because the transistors are never fully switching as such, just slightly charging/discharging the capacitors -- at great expense to power dissipation.
The preferred way is to use the inductor as its own filter, or if that's not possible (again, because of the short L/R time constant), then it can be pre-filtered with an LC stage, and then the LR load connected.
(There is, by the way, a way to use transistors to directly drive capacitors, at high efficiency; but it requires inductors in the supply side instead, and is generally harder to build and operate. I wouldn't recommend.)
Tim