Nigel has provided a comprehensive answer and was very generous to put so much effort and time into it. I agree with his assessment.
I am always tempted to "start another website". However, if I do that I must take my efforts away from websites that are already successful and put that effort into websites that are starting with zero content, zero fans, zero visibility, zero resources of any kind.
If you can afford hire and toss another team of high quality people at improving the sluggish site then maybe you can pull it off, but if your primary website is not dominating the SERPs in your area, then I would probably put that same team to work on the primary site.
I would not start a second website or put resources into a secondary website unless my main website is dominating the SERPs and then I would still be working to raise the bar to fend off anyone out there who is getting any ideas about coming after my turf.