Desaturated - Color management problem / question

My monitor is an eizo cs240 - wide gamut monitor (gamut is set to monitor native which is something larger than Adobe rgb I think - works great for getting good prints from my Epson p800 printer). 

Anyway, the problem is when video editing.  I noticed if I import a jpg from PS or LR they look a lot more saturated in premiere. Hmm I thought.. it got me thinking and then trying to figure out why, which then led me to turn on colour management in premiere. Great - now images match with what I'm seeing in PS and LR. But the video now looks super desaturated. 

So to be clear: in premiere if I select display color management - my video looks washed out / desaturated (compared to how it looked previously in premiere). 

Then when I have exported the video and played back on VLC or Windows media it looks saturated again (same as it did in premiere before the displayplay color management box was ticked - assuming ). 

My question, should I profile my monitor for sRGB when doing video editing? Especially if that's how most people view it (on the web, and likely with uncalibrated kit). In this instance I'd be making changes to my unsaturated look in premiere (on my display) which then would look even more saturated when anyone else viewed it on their machine.