Why do digital modulation when the signal entering the channel is analog?
I think part of your confusion is actually between discrete-time and digital signals. Some textbooks do not clearly distinguish between them.

All signals that exist in the physical world are real. So, the signals that are produced by any transmitter, and the signal that is picked up by the receiver, are real.

In communications, we say we do "digital communications" when the analog transmitted signal conveys "symbols", which are actually numbers drawn from a finite set. This is different than "analog communications", such as AM and FM radio, where the traansmitted information is a (audio) waveform that can take arbitrary values.

In addition to this, these days communications systems are designed as much as possible using digital signal processing techniques. The nature of these techniques is that they deal with discrete-time (sampled) signals. So, a modern software-defined radio receiver will have an analog antenna, a few analog filters and amplifiers, and an analog quadrature downconverter; then, the analog signal is discretized (sampled) and handed off to a digital signal processor for further processing. The digital signal processing can happen on specialized hardware (such as an FPGA or ASIC), or as software running on a DSP or on a general-purpose CPU. This is where digital filters would be used.

Note that a receiver can be implemented using DSP regardless of whether the underlying communications technology is analog (AM, FM, old TV, etc) or digital (BPSK, QPSK, etc).