But therefore, for the same given encoding time, a hardware encoder will give a higher quality output / and / or at a lower bitrate.
No, not really. The hardware encoder cannot
Encoders have "presets", ways of controlling how the encode works, and "levels", what features are available to use in the targeted decoder. Eg, streaming to a STB you might have level 5.1 content, but streaming to a mobile you might have level 3 content.
Software encoders tend to have many presets to determine how much prediction/lookahead to use in encoding a frame. The more lookahead you use, the more efficient the encoding can be, and the smaller each frame can be whilst still encoding the same visual quality. Therefore, in software encoders you can optimise your encode to give the lowest bitrate for the chosen quality. Most videos that are made for distribution are encoded using the preset "ultraslow", because this reduces the file sizes significantly at the expense of a lot of speed.
Consumer hardware encoders don't do this. They have short lookaheads, which keeps the speed high. They use fixed length GOPs, (i-P-B-B-P...), where as x264 will use irregular ones (better quality, better compression). You can't really make it go slower with higher quality per bit (although you can make it go faster with lower quality per bit).