Traditionally, console first person shooters didn't have a save-anywhere save function, because save data could be relatively large considering that until the original XBox came along, every console first person shooter had to save to a memory card (which was either battery-powered, or used some sort of EPROM). These memory cards didn't have a large amount of storage space, compared to the PC's hard-drives of the time, so the games' developers tried to keep down the size of the save game data file by making game-saves either checkpoint only, or making the missions short enough so that they could be completed fairly quickly, and then only allowing saving between missions.
I mean, in a game like Halo or Half-Life, the save-game data must include things like:
- Which level the player is in, and where exactly in that level,
- The players health, weapons carried, ammunition carried, plus any special attributes such as if he has a shield power up (and if so, then how much of that shield power has been used up, or how many seconds until the shield disappears, etc), if he player is in a vehicle, and if so what position is the vehicle in and facing, what speed is it travelling a, is the vehicle damaged, etc,
- The position of every enemy, plus the health of every enemy, the current AI state and awareness of every enemy, and any other enemy-based information, such as if each enemy can have different weapon and different amounts of ammunition, etc,
- Information about which objectives have been completed, what flags have been set, what vehicles are were and in what condition,
and so on.
A good way to minimise the information you need to save is to just save between missions, as then you're mostly just saving the players details and nothing else. You don't even need to save the player's health then, if the player always begins each level with full health.
Failing that, you can make the game only save at certain points, usually when the player cannot go back to the previous playing area, as then the game can safely forget details about the previous area (what corpses are lying where, what places have bullet holes or explosive scorch marks, what enemies are sill alive there, etc), because you can't go back there so it doesn't matter if the game 'forgets' what was in that area.
So console FPSs mostly use either of the two above methods for saving. And it does work well, as long as either the missions aren't too long for games that only save between missions (Perfect Dark, Goldeneye, Timesplitters 2 and 3, etc), or the checkpoints aren't too far apart (such as the Halo games, the Crysis games, etc). But sometimes in a game where you can only save between missions, the missions can seem too long, or too dangerous, which can really harm the gameplay. The only game like this that I can remember offhand is Alien vs. Predator (Rebellion, 2000), but it received an unofficial patch to add the ability to save during gameplay, thankfully.
And some save-only-at-checkpoints games can be very long between some of checkpoints, unfortunately. Turok 2 on the N64 being a good example.
There were some pre-hard drive console games that could save at any point during the game (for example, the PS2 port of Deus Ex), but they were rare.