The Core 2 Duo series is obsolete for CPU and GPU intensive work, but not general usage. That said, these are still useful for basic tasks everyone does despite being so old. You are better off keeping it for a few more years until it is truly obsolete or breaks. Upgrade the memory and hard drive as well so you can further extend their remaining life. If you can afford it, 512GB SSD’s are becoming affordable enough they are no longer out of reach in terms of cost.
As far as an operating system goes, the problem with 10.6 is it began to lose application support when 10.8 came out (and made worse with 10.9), so the situation will be even worse today then it was for me as a 5 year holdout and you will require a more modern release of OS X. It’s really up to you how high you go, but you want to use something with modern application support.
The only downside to having an “obsolete” system is Apple will not service it. However, this is also a partially moot point because the price Apple quotes for repair tends to exceed the value of a used machine with similar specs. While you are on your own for repairs, these iMacs do not suffer from the upgrade problems all of the modern ones do. Even though it’s older (and will struggle in some use cases), it’s easily upgraded and you can easily work on it in terms of replacing most of the components.
I was able to get 5 years out of my T9550 MBP 17” Unibody (Early 2009) before it began to show it’s age and failed beyond economical repair. I kept it so long because that was the second to last 17” ever released - granted the 2011 17” exists but it’s got irreparable GPU issues. You’re stuck with the 2010 if you want a Core i series 17” these days and those are beginning to show their age as well.