For one, they are a nuisance on the road - they really do interrupt the flow of traffic and often get in the way.
How exactly is this specific to trolleybuses? I can think of lots of vehicles interrupting the flow of traffic: Delivery vans parked anarchically, for instance.
Another one is the fact that if one breaks down, it can either a) completely block the flow for other trolley buses or hopefully b) create a large obstacle which other trolley buses have to somehow pass very slowly because their "antlers" have a limited reach, which of course means 2 lanes of road that get clogged.
It looks like Moscow's mayor is a cheapskate who didn't pay a little more to buy trolleybuses equipped with onboard batteries/diesel generators like every modern system has done since the eighties. Well, tough. Anyway, without that oversight, this is a non-issue
With this year's winter, loads of them stalled as well because the lines frosted over, etc.
Again, having no backup system is dumb, plus not performing a defrosting run in the morning (with a trolleybus pole-equipped diesel truck, for instance) is even dumber. Don't blame trolleybuses, blame the dumb people running them.
I will grant you that I'm sure there are ways to minimize these effects that aren't implemented here in Moscow because... well because no one gives a shit what the people have to endure, basically.
I suppose you define “the people” as “the motorists”. I can see each day people served by frequent public transport taking their cars to commute, then complaining that the traffic is unbearable and they've to wake up early to find free parking space. Well, duh! They're just part of the problem.
Trolley buses vs buses in a nutshell:
1) Less reliable
2) Impede traffic
Allow me to point out that the first point is just wrong (electric vehicles are less prone to mechanical failure, not more) and the second, like I said above, is due to bad technical choices. Also, you forget to mention that trolleybuses are more durable (less vibrations than a diesel engine, hence less load on the chassis) and quieter, in addition to being cleaner.
The power lines aren't exactly pretty either =)
Beauty is in the eye of the beholder. I personally like the overhead line's arabesques. Also, a practical advantage is that they advertise the public transport offer to local residents.