Transient analysis of filters inside allowed band
Does it make sense to try to analyze filters in the time domain? Or we
  just accept them as they are (black box approach) and just work with
  them only in the frequency domain?


Analysing them in both the time domain and frequency domain gives more insight and, if designing a filter, it's usually best to do so in both domains. For instance, in the frequency domain it's really easy to see the frequency response but, really not easy to see how the filter performs with transient inputs likes square waves or pulses. In the time domain it's dead easy to see overshoots due to transient inputs (that may of course not be desireable) but quite a bit more difficult to see what the frequency response is - this is why we use the two domains.

It sounds like you are referring to a band-pass filter and if your input sinwave is within the pass-band area then you can totally expect it to look like a sinewave after being filtered.