So what do you guys think, can your typical electronics hobbyist benefit from a spectrum analyzer? Or is this a total overkill (the cost outweighs he learning benefits)?
I like your attitude; "because I want to" is an unassailable reason
But only you can assess the cost/benefit tradeoff.
An SA is a great general-purpose tool for assessing frequency-domain operation, but if you have a specific purpose in mind then there may be alternative options. For filter performance, a network analyser is ideal (essentially an SA plus a tracking generator). Other considerations are what other equipment you already have, and what what else you might want to spend your time (and money) on.
The key technical specs of an SA are its frequency range and its dynamic range. A key "weakness" of some SAs is that the maximum input power must never be exceeded, and that they cannot tolerate
any DC voltage on the input.
As already mentioned, the Digilent Analog Discovery is a very good device within its limitations, and can be used as a network analyser plus other generally useful test equipment.
If you need higher frequencies, then (with limitations) you could use a cheap SDR dongle as an SA, and add a noise source as a poor-mans's tracking generator. See my .sig for more details.
Alternatively, with different limitations, a digitising scope can be used as a spectrum analyser.
My suspicion: your filters are audio or radio-ham frequencies; consider the Analog Discovery if you don't already have a scope and siignal generator.
OTOH, if you are concerned about PCB track impedance, then you will need much higher frequencies. A 1.5GHz SA is sufficient to resolve details ~4cm apart. Consider that a time domain reflectometer is a very good way of directly measuring a track's effect on a digital signal. And due to the duality between the time and frequency domains, it is possible to mathematically transform from one to the other.