Well, that quote from Friedman dates to 2006, way before the info from the wiki article emerged. If I understood the argument, the Type 346 arrays appeared too small to package sufficient power using AESA style S-band Tx/Rx modules built with late 90s, early 2000s technology. Therefore, they found it plausible to assume the radar was closer instead to the X-band Russian Tombstone radar which is quite substantial in size (therefore not AESA?). Given how the 052C inherited the same SAMs from the 051C, the connection with Flap Lid/Tombstone seemed quite obvious at the time. Although the new radar would have to undertake a volume search function in addition to track and fire control.
The Type 346 arrays alone are huge even by today's standards. They only look smaller because the Type 346A arrays are titanic.
As for "sufficient power" why do you need that? The advantage of using S-band is that it loses less power traveling through the atmosphere compared to C-band or X-band. In other words, even if you have a more powerful C-band or X-band radar, the range won't get further because these bands loses more energy as they attenuate through the atmosphere. If you want more power, simply increase the size of the array so you can add more elements. Each module has a power ceiling anyway because Gallium Arsenide as a substance has a voltage breakdown limit anyway, and in order to get more power for each element, you transition to the use of Gallium Nitride that has a much higher voltage breakdown limit, By then that's 2015 technology at least. If heat is a problem due to the high power density, then you deal with it via cooling. Liquid cooling, like the Type 346A, will allow for more power per module than the air cooled Type 346.
You said that the S-band elements would have to be at least half the size of its bandwidth. What is the upper limit?
S-band is 7.5 cm to 15 cm in frequency. C-band is 7.5 to 3.75cm. X-band is 3.75 cm to 2.5 cm. The smaller elements would pose a greater technological challenge due to the miniaturization required.
As for Tombstone, if you're a member of CDF I made a long explanation of what type of phase array Flap Lid is in the PLAN Navy radars thread. Just the gist of it. There are four different kinds of phase arrays or PESA. The kind you see in Flap Lid, Tomb Stone, MPQ-53 and even the Chinese HT-233 is called a spaced phase array. Each element does not have a line feed from a central amp, nor does each element has a transmitter. Each element consists only of a receiver and phase shifter. The whole array acts like a lens, there is a large emitter on the back with an optical lens and four horns (monopulse feeds) that would project the beam to the array. You only need to power the array for the phase shifters, not for transmission.
Spaced Phase Array is a very different technology from AESA, where each element has a separate amp and TX emitter on its own.
PESA like SPY-1 or Russian BARS on the Su-30, is similar to an AESA where each element has its own TX emitter, but each emitter is connected via parallel (same length) line feeds to a single large amp (TWT, Klystron). You may refer to this as a parallel line feed phase array.
The other two phase array types is the frequency scan or serial line feed phase array, like Fregat radars, and the Reflective Spaced Phase Array, which is a rare type, as it works like the spaced phase array but instead of refraction, it works on reflection with the feeder in front of the array.
The Tombstone/Flaplid -> Type 346 does not succeed not just because its a spaced phase array vs. AESA, but because the Type 346 is a search radar and Flap Lid is a fire control radar. On the 051C, there is an S-band Fregat radar that acts as the main search radar. But on the 052C/D, there is no Fregat search radar, which points to something else acting as the ship's main search radar. That only leaves you with the Type 346 itself.
Search radars generally work on the L or S-band. Fire control radars on the C or X-band. X or C band can be recruited as search radars if a dedicated S-band radar is lacking but won't be as efficient. There are a few cases where warships don't have S-bands, the vast majority has S-bands, and every PLAN warship has an S-band. It would be absolutely weird if the Type 052C/D doesn't have an S-band when even Jiangweis do. The shape and location of the Type 346 arrays are clearly inspired from the SPY-1 radars, and the SPY-1 is S-band.