How powerful of a CPU do I need if I'm not gaming?

I've been working on building a new computer for my church. I have been purchasing parts off r/buildapcsales toward that goal. ($10 PSU, $20 case)

I'm wondering how powerful of a CPU I need? Considering I will be running off integrated graphics and will not be using a dedicated graphics card. The computer I am replacing is only used to display HD video and run Libraoffice impress.

The computer I am replacing struggles significantly with HD video and has a significant lag when using LibreOffice especially when using slide transitions in that software.

I do not intend to install Windows on this new computer instead I prefer to install a flavor of Linux. I am uncertain if this will affect the processor recommendations.

My question is this which processor would be adequate for these use cases? It doesn't need to be the current generation of processor either. I don't mind purchasing used and refurbished equipment if there's a particular good site to use.