Totally depends on the CPU load of your game and the browser sources, that you use.
In some games, my 1700x @ 3.8GHz can barely run x264 1080p60 with very fast preset, otherwise the amount of encoding lag is noticable.
The 2600 has slightly better "per core" performance than my 1700x, but 2 cores (4 threads) less, so when it comes to encoding, I think the 2600 will struggle to encode x264 live in 1080p60 while playing games that use 4 or more cores.
720p60 on the other hand is no problem with medium or even slow preset.