Thanks for your time and effort! I just ran some tests myself.
I created a new freeplay save and changed the bot speed value to 2 by accessing the mod settings from the map generation menu. Then I saved the game as "test-1". After this I restarted Factorio and checked the mod settings from the main menu (Settings > Mod settings). The bot speed factor was still set to 2.
For the second save, I changed the mod settings directly from the main menu. I set it to 4, then I created a new game, saved it as "test-2", and restarted Factorio. As I expected, the value was still at 4 while accessing the mod settings from the main menu again.
My third test seemed to reproduce your issue, where I created a new game first, and then changed the bot speed to 6 from the pause menu (in-game). I saved the game as "test-3" and restarted Factorio. This time, while accessing the mod settings from the main menu, it was still at 4. However, the values were correct while loading all three of the saves: in "test-1" it was set to 2, in "test-2" it was set to 4 and in "test-3" it was set to 6. This is the intended behavior.
Let me explain it further. This is not a bug with the Speed Settings (probably not even a bug at all). The mod only ever reads from the settings (it never writes to it). The rest is handled by Factorio itself. Also, those settings are not global for all of your saves; they're local to the individual saves. Factorio just seems to remember the last used values if they were changed before the map generation. It doesn't seem to do so while they're changed in-game (after map generation). It's probably just intended behavior, but only the Factorio devs can answer that.
I hope this helps.