The 'solid state' coils are essentially regular encapsulated coils, but with a case around them.
'Coil While' is due to current passing through coils causing them to heat up, and the heated coil expands. Current then reduces, the coil cools and contracts. Do this many times per second, and you have a tiny speaker. The high cycle rate (frequency) means the coils emit a high pitches tone: the 'whine'.
The entire reason coils are on the board is part of the power delivery circuitry. They are fed AC at high frequencies, and are part of the system that converts that to DC for delivery to the GPU*.
The only ways to eliminate coil whine, no matter what encapsulation is used, is to either eliminate the expansion/contraction, or to change the frequency at which it occurs to move it out of the audible range. Changing the frequency is trickier than it sounds because the frequency varies (as GPU load varies, which is why coil whine a a weird varying screech rather than a pure tone), as frequencies get higher you start to have even worse RF leakage issues, and even at high vibration frequencies you can still get audible harmonics.
You could make the coils out of Inconel, or another superalloy that has very little thermal expansion. However, these materials are not cheap, and may not even be suitable for construction of inductors (I can't find any suppliers of SMT Inconel inductors).
* long answer: 12VDC is received by the card, MOSFETs chop that up at several kHz into 12V pulses with around a 12:1 duty cycle, coils shape those pulses into a sine-like waveform, capacitors buffer that waveform into a smooth output. Because of that 12:1 duty cycle, the final DC voltage is 1/12th of the input voltage, so you have a DC conversion of 12V in to ~1V out, perfect for feeding the GPU. This is a super-simplified description for how a switched-mode DC converter works. While there are other DC-DC converter designs, switched-mode converts are best at handling the high powers required efficiently and in a compact form factor.
