For reference, a typical schematic can be found on page 13 of this datasheet (similar but unrelated part); or on numerous appnotes, or schematic collection sites.
https://www.onsemi.com/download/data-sheet/pdf/fsq0370rla-d.pdf
In this case, it appears there is an inlet fuse, MOV ("TNR"), inrush thermistor / NTC, X cap, common-mode choke, then FWB (D8) and one electrolytic capacitor (with no additional filtering components near it -- L1-R2 and L3-R8 are shown in parallel so that one can choose which components should be installed; basically, this would be the simplified form, jumpering them out, since the CMC and stuff handle filtering on the AC side instead). Likely the particulars of the control output circuits differ as well, but the major components will be common.
The most likely components to fail are electrolytic capacitors (they dry out over time, particularly when hot), and semiconductors (due to exceeding ratings, or some aging failures). MOVs can also wear (but it looks fine from here, clean bright blue?).
Often, semiconductors fail as a cascade mechanism, for example the switch IC is working harder because the capacitors are failing, and eventually its ratings are exceeded and it dies; the fuse is taken out, but in the process, a good thousand amperes or so rushes through the rectifier, transformer, etc., so those components are at risk as well. If using a separate controller IC (rather than the integrated kind here), that may be destroyed, along with related resistors. Sometimes the output side capacitors or diode fail, and the resulting poor load (excess peak voltage, or a short circuit condition) stresses the power switch to failure.
The component in question is a 0.1µF 10% 310VAC MKP, metallized polypropylene film capacitor. As you can guess from the tolerance, its value should be 0.09-0.11µF. Clearly out of spec and in need of replacement.
X type capacitors are generally designed to fail open, by a process called "self healing". This gradually degrades the value (lower C, higher ESR) until it ceases to be useful. Likely this has happened here. Healing occurs in response to excessive voltages such as mains surge, which is expected to occur throughout the life of a product, making this component a wear item as well.
These can also fail simply by oxidation: the metallization is extremely thin (~100nm!), and mere O2 / H2O diffusion into the capacitor gradually oxidizes it (aluminum into its non-conductive oxide), resulting in the same sort of failure mechanism, without any "healing" events occurring at all. (This typically takes over a decade to occur.)
Would it work without? Perhaps. This capacitor, along with the CMC and any related components, filter out switching noise from the circuit, before it reaches the mains network (lots of long wires -- one heck of an antenna). Maybe the unit itself would operate (clearly it's capable of running in the presence of its internal noise, though the presence of this capacitor does affect that a bit, still), but maybe connected or nearby equipment wouldn't. Maybe radio reception would be knocked out, maybe on bands you aren't even aware of (there are licensed, commercial and emergency bands, for various purposes, all across the dial!).
