Using vestigial sideband instead of double sideband supressed carrier could make the baseband stereo signal more compact.
Would there be any other benefits or downsides of employing VSB?
The downside would be that the receiver would be more complicated. At the time that FM stereo was introduced, receivers were built entirely from discrete electronic components, and costs were directly related to the number of components required.
Today, of course, you can add arbitrary numbers of transistors to an IC essentially "for free", so there are fewer constraints on complexity, which is why we now have robust multichannel digital broadcast in the same bandwidth.
There's another reason that DSB-SC was chosen, and it relates to why you sometimes hear the term "multiplex stereo" in association with it.
There are two ways to decode an FM stereo broadcast. The first is to demodulate the baseband signal, which contains the L + R "sum" of the two stereo channels, and separately demodulate the DSB subcarrier, which contains the L - R "difference" between the two channels. Using an analog summing matrix, you can then re-create the original L and R discrete channels.
However, if you've ever looked at the raw signal coming out of the FM discriminator, containing both the baseband and subcarrier signals, you'll notice that it appears to be a time-domain multiplexed version of the L and R channels. You can decode it by creating a 38 kHz clock from the 19 kHz "pilot" tone that is also in the raw signal, and using this clock to directly "sample" the discrete L and R channels out of this composite signal.
(Image taken from this site, which has more details on this process.)
This is also why many early FM stereo receivers had a "Phase" control knob — this directly adjusted the phase of the 38 kHz clock relative to the pilot tone for the best stereo separation.