when I'm selecting under Format: Computer for Export File, I have only the option for the H.264 Faster Encode option, the better Quality Option is missing. Does anyone has a clue why the option is missing? Thank you for your help
Probably, but it's only a bug if it wasn't intentional. Likely they would have removed the whole line if it was intended.
In recent versions I can't discern a visual difference between the faster vs better quality Computer>H264 exports, so maybe they decided to drop that option. But as you said, why leave the "faster encode" wording? IF it's a bug I can't believe that was missed. That export option must be well-covered by the test matrix, plus even ad-hoc testing should have caught it. My former product manager at a major software company would have lambasted us if we let something like that out (if a bug).
On my iMac Pro running Big Sur 11.6, I tested FCP 10.6 export performance using UHD 4k/23.98 optimized media. I couldn't see any difference in quality or export time between Computer>H264>Fast vs Video/Audio>H264. They were both about 58 sec to export a 120 sec timeline with no Fx.
However the Computer>H264>Fast file was encoded with a GOP length of 12 vs the Video/Audio>H264 had a GOP length of 30. File sizes were the same. If re-imported the file with a shorter GOP length was a little smoother to edit in FCP and Resolve Studio 17.4.
I then exported the same source material from Resolve Studio 17.4 to UHD 4k/23 H264 (with all files rendered to ProRes 422 cache before export) and on the default case it was slightly faster than FCP but the export file had a GOP length of 120. If reimported it was very sluggish to edit in either Resolve or FCP, at least on my iMac Pro.
In theory a file with a longer GOP length will be more compressed but that's also determined by encoded bit rate. The two FCP-generated files were about the same size and used about 50 megabit/sec. The Resolve-exported file was about 25% larger and used a bit rate of 66 megabit/sec.
I then changed the Resolve GOP length to 12 and the export became about 3x slower than FCP. They must have some optimized code path for the default case, and if you stray from that it gets slow -- at least on my hardware.
Regardless of whether the FCP 10.6 H264 export dialog for "Computer>H264>Fast" is a bug or intentional, it works fine and produces good quality files.