The Spirit of Fire was, even within the story, slated to lose extremely fast against the Banished. In fact, the only reason it persisted as long as it did was due to Atriox likely trying to capture the ship mostly intact rather then simply annihilate it. The problem is that covertly boarding and destroying Covenant ships/fleets is literally the bread and butter of S-IIs, and the Spirit of Fire has three. Without the Enduring Conviction, it scarcely matters how much of a numerical advantage the Banished have, their wings are clipped in a more literal sense.
Another important factor is force concentration. If Atriox is has 10x the numbers, even 100x, but is trying to occupy an extremely large area, it thins them out considerably. Battles are not fought with the full might of both sides at the same time. The Spirit of Fire can win individual battles by defeating the Banished in detail. What that means is that they focus a considerably larger proportion of their force on a singular point. Say the Spirit of Fire has 4,000 soldiers and the Banished have 40,000. If the Banished are trying to hold 100 different locations, their average force concentration would be ~400 soldiers per location. Obviously this is just a basic average, more critical areas will have vastly more while less important areas vastly less. What this means is the Spirit of Fire, by only focusing on a handful of missions/areas at a time, can match or exceed the force concentration of the Banished. This advantage is further amplified in the wake of the Enduring Conviction's destruction, as it becomes vastly harder for Atriox to predict UNSC deployments with no limitation or pressure on where the Spirit of Fire can operate. Not to mention the advantage of orbital superiority in general, the defensive safety and the ability to have heavy fire support on demand. Red Team themselves are massive force multipliers in and of themselves. Individually, with very little in the way of direct support, they can frequently take on 1:100 odds against Covenant/Banished combined arms and win decisively repeatedly. James in First Strike says as much of their capability, and the range extends up to 1:500 odds if they have preparation and limited support (And is further supported by many of the games, where you as a lone Spartan kill your way through 100-200 Covenant in the span of 15-30 minutes). That means Red Team, deployed alone or with limited support, is essentially 300-1,500+ dead Banished with few, if any UNSC casualties. That's an insane advantage to have when it comes to facing an opponent far greater than you numerically, but distributed over a vast area and much more limited in mobility.
Also remember that the Ark is huge. The surface area is vastly larger than Earth's. We're talking travel distances likely in the tens of thousands of kilometers, and the potential for some over 100,000km (Diameter of the Ark being 127,000km or so). The Spirit of Fire can traverse these distances extremely rapidly, as can deployed craft that don't have to contend with atmospheric drag. The Banished on the other hand are limited to transportation via Phantoms, Spirits, and Liches in atmosphere, as they would likely be easy pickings of the Spirit of Fire should they attempt approaching an active combat zone from orbit.
In summation, mostly due to the loss of the Enduring Conviction, the immense mobility advantage and near intangibility of the Spirit of Fire allows them to continue their survival and operations on the Ark, slowly bleeding the Banished with strategic concentrations of force.