Pausing a game is not as simple as freezing its entire state (and saving is also not as easy as dumping the entire game state to disc, at the least you'd end up with gigabyte-sized savegames).

Many game systems need to remain active in paused state (the UI needs to remain working for example, and actions in the UI may also manipulate game state (inventory, crafting, levelling up...). There are also plenty of games with 'active-pause' where the user can issue commands to units while in paused state.

Likely more difficult.

Live migration boils down to copy memory over the network, stream the page faults till you converge enough, and resume execution on the other host. It’s not a hard problem but a precise and tedious one.

Pausing a game might involve a lot of GPU contexts to freeze, network resources to pause, storage streams to pause, input handling, sound, etc. Add to that physics engine that may be tied deeply in the system and you end up with a hard problem.

What a VM does is not the role of the hypervisor, thus it can apply its hammer that works in pretty much all cases, and VMs are pretty much all the same. On the other hand, all games are bespoke with custom plugins and custom integrations, which make them the opposite of "generic pause implementation".