It's always important to keep in mind that more than 80% of signal processing is filtering. However, not every filtering operation should be "viewed" in the same way.
Let us take the example you mention, of an "ideal known channel" (assuming it's not in deep fade). In wireless communications that would mean perfect knowledge of channel to the receiver. Typically, you could invert the channel in a very basic and often used scheme known as Zero forcing i.e. if $$y=hx + n$$ where y is received signal then the equalization is simply $y/h$ assuming real signals and noise. Similarly an MMSE equalizer ( optimal for most SNRs) has a typical mathematical expression that you would find in standard textbooks. The point is it would be less intuitive to look at this equalizer from the perspective of it's frequency response. It might be more intuitive to look at it from the perspective of vector spaces in this case. Especially as the degree of freedom of a network grow. For ex: a MIMO system.
Filtering is essential to iradicate BER, simply because if you don't filter ( as mentioned above), the time domain signal has been essentially "changed" by the channel.
We are always trying "mitigate" the effects of the channel. I would say "mitigate" rather than "invert" because it is sometimes not even possible or doesnt make sense to simply invert. For ex: in the scheme above $y/h$, notice that the noise is also devided by $h$. If $h$ is small, then you boost the noise as well. So in such a case a MMSE equalizer would make sense. When viewed in line of probability theory, the MMSE equalizer is calculating the expected value of the posterior distribution of $x$ (your signal) after observing $y$. So in that sense it is not inverting the channel.
Long story short, we try to do what's best for the system depending on bit rates, how close the bit rate is to channel capacity, SNR, modulation schemes etc. Rather than simply invert the channel.
There are other scenarios such as an antialsiaing filter, in such cases it is best viewed from it's frequency response, like cut off frqeuency and phase responses. You would get a better idea of the time domain filtered sequence looking at it from the frequency domain perspective. For ex: the cut off frqeuency would tell us what maximum gradients are tolerated in the input signal to the filter and hence are part of the filtered sequence or are smoothen out. The phase response would tell us which freqencies of the filtered sequence, would be delayed by what amount. Whether delay would be constant for the whole signal (linear phase) or there would be different group delays (non linear phase).
Indeed the filtering will change the shape of the input signal, otherwise usually nothing interesting is happening, but the filter itslef is not designed to "distort" the signal, it is designed to "better" the signal according to application, indeed the shape /magnitude of time domain filtered signal will change after filtering, but for the better of the system.
