Can a computer game can run on a boot mode without the aid of os?
Yes, it is possible to boot computer to the specially designed program, which will be some game. But this will be harder, because there will be no drivers for advanced hardware, only elementary BIOS functions, basic video memory access. For example, if author want to add sound to the game, he can used PC speaker, or write own driver for ... for which card? Sound blaster, or RTL? What if user has another card? Same problem for video: game may know about some Radeon and will not work on older or newer AMD card or on Nvidia or Intel.

So, writing a bootable computer game is the same as writing both the game and the Operating System with drivers (or stealing its codes like in PS4's Orbis OS with FreeBSD sources or SteamOS with linux sources).