1080p@60 is “only” around 500MB/s, which should have been possible a decade ago. PCIe 1.0 x16 bandwidth maxed out at 4GB/s, so even if you weren’t on a top of the line system with PCIe 2.0 (or brand new 3.0!) you should have been fine on that front[1].
More than likely the CPU wasn’t able to keep up. The pipeline was likely generating a frame, storing it to memory, copying from memory to the PCIe device memory, displaying the frame, then generating the next frame. It wouldn’t surprise me if a ~2010 era CPU struggled doing so.
[1] Pretty much any GPU’s memory bandwidth is going to be limited by link speed. An 8800GTS 320MB from 2007 had a theoretical memory bandwidth of around 64GB/s, for reference.