I'm running an Athlon 64 X2 6400+ with 4GB RAM as a Web Server, with Ubuntu as the OS. The 6400+ is the fastest CPU my motherboard will accept and for my needs it's absolutely fine. Still usable as a modern PC, though I would hate to have to run it as my main day to day system. Even an Intel Core 2 Quad from the same era feels the strain on the modern Web. I used to leave Firefox open with 100s of tabs on a Core 2 Quad Q6600 with 4GB RAM and the Firefox process would need ending every couple of hours as it bogged down the system. 🤣
I've actually recently retired a PC from active daily usage that had a slower CPU than you're aiming for - Athlon 64 X2 5200+ with 4GB RAM, running Windows 10. It wasn't a speed demon, but served my family's needs well. Only the occasional moans about slowness and it was pitiful when they tried to watch streaming video on certain websites. The 6000+ should cope a little better there, but certainly with lots of 100% CPU usage spikes.
I also have several other Socket AM2 based systems, all still useful as modern systems in some way:
Athlon 64 X2 5600+ that's in use as a HTPC, running Windows 8.1 with Kodi. Handles HD video playback & live TV etc with ease.
Athlon 64 X2 5000+ that's my classic Hackintosh setup, to cover early Mac OS X era games & apps on Mac OS X 10.6 Snow Leopard. Been playing Doom 3 on that setup recently with max settings.
Sempron 3500+ this is currently just a spare PC that I may make into a backup HTPC. I did overclock this from 2.0GHz to 2.7GHz though, which certainly helps.