That's respectable, I'm sure we didn't mean to undermine you. My mistake to assume you were a beginner.
Anyway, what I'm hearing is that you just want suggestions on further bike parts? Here's just some popular high(er) end parts that are common:
Frame: I'm sure you will enjoy the Leader eqnx. Keep in mind that it has a trail of 62mm, which will handle significantly different to your pure fix. When steering, it will feel slower and more sluggish (but be more stable when going fast). There will be toe overlap and obviously, a much more aggressive geometry. It will be a learning experience-- a fun one at that assuming financial means is not a barrier.
Stem: various 3t stems depending on your choice of angle, various Thomson stems (X2, X4 depending on choice of angle). It would be best to choose your stem length based on the geometry of your frame to compensate, but its fairly safe to just get a 80mm or so. Angle of the stem is up to you.
Seatpost: Thomsons are solid. They're a very popular choice.
Handlebars: this can highly vary. be sure to get the appropriate clamp size for your stem, whatever that may be depending on your choices. oversize (31.8mm) stems are the standard.
Saddle: this can highly vary as well, and should be properly fitted with your sit-bone width.
Wheelset: The entire H Plus Son line is exceptionally well crafted. You can't go wrong with any of its wheels. I have the archetypes and i love them
Crankset: Yes, omniums are good. They will be good for street riding as well.
Pedals: It is your decision to go clipless, toe-clips, or platform. All three will be fine if you will be riding single-speed with brakes. A popular choice for toe-clips are the MKS GR9 pedals with All City Clips.
Brakes: I'm not too familiar with single-speed dedicated brakes. I have Shimano 105 calipers with tektro cross brakes. Perhaps you want to try hoods. Keep in mind many modern road bike brake levers are "brifters", brake levers and shifters. Such as if you purchase the Shimano 105 brake levers, it will also be a shifter. Many companies such as tektro make brake levers only for the hoods meant for single-speed and non-STI/other integrated road bikes.
Tires: If you will not be skidding, anything decent will be fine. Continental is a great brand.
This is just my 2 cents with my limited knowledge of bikes and I hope I have addressed your concerns. I hope I have helped, and if you have any questions feel free to ask. After all, we're all here to establish a community about bikes-- not to judge and exclude.