For 4k work you will need to double or triple that budget - roughly $1500 to $2000 - just to edit comfortably at all at that resolution. A $700 machine, no matter what age, can only comfortably handle 720p at best.
And this is because of the lack of a truly competitive CPU offering from AMD for many years, and Intel was not motivated to improve performance significantly while keeping prices to a minimum. As a result, the bang for the buck in CPUs has not gotten better, and in some cases actually worse, than it was a few years ago.
