Single phase motors have been known to put the brightest, most electrical savvy people in their place.
Named for how they electrically start, (shaded pole; permanent split-capacitor run, split phase, capacitor start, and so on...)
these motors have numerous (necessary) components required for them to get moving and run.
Capacitor (s)
Internal Stationary Start Switch (s)
Shaft mounted Centrifugal Mechanism
All of the above components (including some not listed), link a chain of functioning events required for the motor to start.
Single phase motor exploded view URLs found via Google for reference.
Most single phase motor troubleshooting efforts begin with what can be checked externally on the motor before opening it up.
As already mentioned, the capacitor(s) are a good place to start checking first, followed by a quick mechanical inspection.
While there is a useful lifetime to capacitors, and they do indeed fail due to age... a bad capacitor is often attributed to
a failure in one of the other component "links" including the driven load.
In your case, we'll rule out any mechanical issue with the driven load because it appears to turn freely.
So; if you find the capacitor to be bad, you have to also inspect the stationary switch, and shaft mounted rotating
switch (Centrifugal Mechanism) to determine if either of these played a role in the capacitor testing [no good].
Stationary switch contacts wear out from age, become welded together, or can be obstructed from operating properly by debris inside the motor.
The insulated tabs on these switches are subject to wear as well.
Hence, a malfunctioning switch can keep a capacitor in the motor start mode causing it to electrically fail.
In addition, the start winding itself can electrically fail due to any of these other start mode components failing.
A single phase motor with a failed capacitor will typically exhibit some shaft turning during the energizing of it under no load.
And yes, giving the motor shaft a "spin" by hand can often get it started.
The outright "buzzing" you're referring to would tend to point toward another investigative path to take.
Mechanical Inspection.
With the motor assembled, attempt to move the output shaft by hand up-n-down or side-to-side to determine if the bearings, or bearing housings are loose.
Out of tolerance / loose bearing housings allow the rotor to be pulled to one side when the motor is energized, which prohibits it from starting.
Noting that this is a belt drive application, typically the opposite drive end motor bearing pocket will wear out first over time due to excessive belt tension.
The front bearing acts as a pivot point as the rear housing wears out. Yet! There's also the condition of the front bearing journal wearing out first...
allowing the rotor to move side-to-side during the starting process.
In rare cases, an open rotor bar will not allow the motor to start, but that scenario has additional failure attributes that are likely
not the cause of your motor not working.
So having written somewhat of a mini essay for a Sunday morning... the answer to your question of,
"... whether the motor needs servicing (if it is indeed serviceable) or if it is more likely to be the big capacitor
connected to the motor which may be at fault?", is Yes.
What you may likely find though is that the cost of troubleshooting such a motor is not worth the expended
expertise required in determining the exact cause of it not working, and you're perhaps better off considering outright replacement
of the item if it's even available.
Single phase motors can be a "kicker". Especially today... due to their "Throw away, and buy new factor".
Sorry about the long-winded answer.
Always enjoying the forum,
John