For the Infinity keep it a mystery. Fully corrupted data logs, excessive damage, and clear malfunctioning slipspace drive. There is only one to be found and the mystery of how and when it got there is the hook.
They can get it working but they can never get a full answer only speculation and theories.
The first few things the UNSC would have done after stabilizing on New Harvest is sent up a flash clone facility and some sort of terraforming infrastructure.
There should be expansion on the interaction between the Hegemony and UNSC that led to the issues they have at the start. As well as how long they have been in the Sector.
I say a few months, to long and the UNSC builds up modern fleets that can wipe anything, to short and their to weak to be interesting. I have a few Ideas and it brings in the iron shell to assist with that, and of course, they know the name "The Immortal" but nothing else except the ship was for unknown reasons scuttled, except the charges never blew and the ship drifted endlessly waiting for a new crew.
A few months is nowhere near enough for the UNSC to do anything. They are a battlegroup with a single colony fleet. Cut off from any support building on pretty *** world from scratch. Considering the planet is still a Frozen one their original target wouldn't have required much if any terraforming so they didn't bring much of the equipment with them.
It also isn't enough time to build or repair anything. Unlike the rest of the factions the UNSC didn't have access to nanoforge technology to build their equipment. They built it the old fashioned way by hand. The mainstay of the fleet is makeshift shield equipped pre-war ships. Even in the Halo universe the post war ships rarely had shields or all the upgrades because of limited production ability.
It would take years for the UNSC to even stabilize. Throw in the isolation from no one helping them because they use AI beyond the restrictions of the Domain.
As for the Infinity. The name Immortal is bad but not any worse than the actual names used (Infinity, Eternity) but that should be scrubbed as well. Don't make it a failed scuttling job because that has to imply that the crew are around somewhere in the Sector. The Infinity is a strong ship but it can be overwhelmed so it being damaged in fights to the point a escape into slipspace caused this version to show up in the Sector makes sense and leaves it vague enough to work with any future lore.
Characters and Officers in game should be telling their stories. There also should be a core fleet flying around the system like what the Iron Shell and Mayasuran Navy has.