The heaters are at least partial reason, yes. A cold filament will draw more current but the current is limited when the filament begins to heat up and due to that increase in resistance. Basically, the operation of it is very much like that of a light bulb limiter.Tube amps have an initial startup surge because of the heaters correct?
BUT who says solid-state amps have only fast blow fuses...? I've seen plenty with slo-blo and for the good reason that the inrush surge can be ludicrously high in them as well. They don't have filaments alright, but the reservoir capacitances are often substantially larger than in tube amps and the rectifiers are almost without question solid-state and "instantly conducting". Also, throw in a toroid with high VA rating and the stage is set for massive inrush current draw.
It's not so much a problem of < 50W amps but once you start to get past that the inrush is a very real phenomenon and will eat conservatively rated "fast" fuses for breakfast. Either you throw in a fast fuse rated for a very high current, or a slow blow fuse with more effective rating, closer to limits of real current draw of the amp when voltages have settled.
Some SS amps do have inrush current limiters, mainly PA amps and alike with substantially high output power ratings. I think the rule of thumb recommendation is to start fitting those things in once the VA rating of the power transformer goes past 300. But it's somewhat rare to see amps in 50W - 200W range using such. There are thermistors in eg. Fender and Crate amps, but then again eg. Marshall SS amps often seem to be without inrush limiters and are fused with slo-blo. If it says something like "T1A" or "T2A" in the schematic it refers to a slo-blo. (T = time lag). And it's quite common to see that "T" in there.
I would think that without the thermistor or some other kind of inrush limiter almost all of those SS amps with fast-acting fuses would indeed require a slo-blo mains fuse, since the rail-to-rail voltage is usually in the range of 60 volts or more, total rail capacitance can easily be in the range of 10000uF, VA rating can be close to 200 and the rectifiers are semiconductor diodes.
I think some tube amps with tube rectified PS could infact cope with fast blow mains fuse since the rectifier limits the inrush and sets a limit for the reservoir capacitance.