Since you can install games on a flash drive on an Xbox 360, and flash drives generally have faster random reads than hard drives, would it give Xbox 360 games better performance by installing them to a (fast) flash drive instead of the traditional mechanical hard drive?
To the best of my knowledge, no. I haven't tested it myself, but I do know someone who didn't have a hard drive for sometime and relied on a USB drive. His stuff loaded significantly slower compared to when he finally got a hard drive. As mentioned in the comments to your question, USB 2.0 is likely to be slower compared to your Xbox's hard drive.
Now this isn't my personal experience, but from what I've read and heard from others, I believe your hard drive is the way to go :)
First, consider what flash drives are fast - some flash drives have downright low write speeds with a maximum of 2MB/s.
The Xbox 360 USB 2.0 interface goes up to 35MB/s out of the USB 2.0 port, generally allowing up to a 60MB/s for PC's, so it's even more of a bottleneck for the console restriction than the interface limit itself. But still, you mentioned 15MB/s is mandatory for flash drives, since that is the Xbox 360 DVD Drive speed. Finding a flash drive that goes up to that is a bit difficult in and of itself.
I was just lucky when I bought my 32GB Sandisk USB 3.0 drive, and it just so happens to write at 15MB/s on my laptop 2.0 port. Usually, 3.0 flash drives make way better use of the 2.0 speeds.
I'd recommend the HyperX 3.0 since that actually achieves 30MB/s on 2.0 ports, which is near the Xbox 360 USB limit.
Yes it does, trust me, I downloaded Skyrim legendary edition on my 6.9 flashdrive connected to my Xbox and it is so much faster! Dude I promise you, if you do it, the game will run like brand new. You will still need the disc to start the game but after you will notice that the disc has stopped spinning, the game will run smoother then ever. Trust me I have tested this and the results amazed me.
I just installed Skyrim on a flash drive instead of the disc and the performance difference is great! For instance, when creating my character at the beginning, running the disc, the menus ran like a DOG. Waiting 20 seconds or so just to view each different race was infuriating. I thought my 360 had frozen or something was wrong with the game. From the flash drive, the menus were perfectly responsive, snapping through the choices instantly. Also, in gameplay I quickly noticed that hiccups in the frame rate disappeared. On Assassins Creed IV Black Flag, the difference was startling. The frame rate seemed to double at times and the graphics improved as if my 720p resolution jumped up to 1080p. It made those games more fun. Instead of compromising and having to just accept and try to overlook the performance issues from the disc, the flash drive installation cleaned it ALL up.