How important is a Linear Power Supply Versus a Switcyed one?

Both are switching producing high frequency noise, but SMPS done poorly can be really bad, much worse than linear PS. On the other hand SMPS can be much quieter than linear PS if designed for low noise (zero current, zero voltage switching). Jeff Rowland uses SMPS even in preamps, where power losses are insignificant (where he used batteries before). Benchmark reduced noise by 10dB in DAC2 by using SMPS instead of linear PS, they used in DAC1. They also claim that such low level of noise wouldn’t be even obtainable with linear PS. If Sutherland offers this option, then linear PS is probably better, but I would look for reviews, or check return policy. It is always possible that they offer this option only because people demand it (SMPS has bad rap from crude, very noisy computer PS).

I think this is a mod, not an external box, so return would be messy if not impossible. The question is whether it would be noticeably better. 
Forgot to mention that I have a Furman Elite 15 conditioner which got rid of a very slight buzz I had with the volume turned up with nothing playing. Does this make a difference in this discussion/decision?

you know the answer... depends on what unit

some will see a drastic improvement, some slim to none, depends on how much internal filtering/noise isolation is happening inside the unit

but usually it matters and improves at least a teeny bit

That's what I am afraid of....  a teeny bit. I don't consider myself a tweaker and generally don't do things like aftermarket tonearm counterweights, cable stands so they don't touch the floor or that type of stuff. I do care very much about quality and was very pleasantly surprised by a tonearm I bought last year that blew me away (the phono stage made almost the same difference) but a 2 step turntable trade in/upgrade that cost more than the arm made only a very subtle difference and I was disappointed with that, but liked a couple new features so I kept it.  

I am very much aware of the law of diminishing returns and I think I am at the point where my strategy of paying up to $1000 for something noticeable being a no brainer is now impossible. Maybe I am wrong here. I think I am going to call Ron Sutherland - he is a very nice down to earth guy.

I have always found a linear power to be an improvement over a switching power supply.  This is both in analog sources/preamp as well as digital transport sources.  If your system is very warm and laid back, you may not notice any difference.  I think your Plinius integrated is going to be a much higher resolution and revealing amp.  Upgrading your phono stage to linear power supply will likely remove some glare from the sound and give you a smoother and more realistic sound.  I took a look at the Sutherland Insight phono stage.  The power supply is a separate board that can easily be taken out and replaced.  If you google for picture of "Sutherland Loco", you can see what the linear power supply board looks like.  They would use the same one in your Sutherland Insight phono stage.