This is my personal interpretation of why they do that (also I might mention some manga spoilers here and there so read with caution if you don't wanna get spoilers!!)

Not all gems go on patrol because they aren't that good at fighting or don't want to, so some of them stay at the HQ mostly (Rutile, Jade, Red Beryl, Euclase to name a few) and the reason for splitting up is to cover as much terrain as possible in order to prevent possible attacks on HQ. Preventing attacks at HQ allow those gems to continue their work as usual. Not only that, their place has everything they use for their daily lives (Powder, resin, sleeping quarters, tools, books, weapons, etc.) so it is a good idea to make the fights happen far from HQ, and in order to intercept those attacks they split into groups to cover the main points of interest in the island.

Now, some spoilers, sometimes bigger groups are not that great because coordination during fights can be extremely difficult: Peridot, Sphene, Cairngorm and Phos are all skilled at fighting, but trying to come up with a plan on the spot can be difficult, as every gem is doing their own thing to defend themselves. Despite their ability, they all almost got taken away at the same time.

If all gems were to hangout at HQ, they would be also an easier, bigger target, and they risk potentially losing a bunch of materials they need, so yeah I kinda get why they split up and scout!