Transmission line considered is lossless. Basically the configuration is simple.
For transmitter side : a non ideal source (can be voltage current or power) feeding a transmission line (TL1) with source impedance (Rs) not matched with transmission line and transmission line terminated with antenna with unmatched total impedance (Ra1) with respect to transmission line.
For receiver side : another antenna (Ra2) feeding transmission line (TL2) which in turns feed a load resistance (RL) here transmission line is unmatched to either antenna and load.
What I think is that as reflection of wave is internal to transmission line the power transmitted should be found using Zinput of TL1 taking Ra1 as load impedance. A part of voltage will drop at Rs and rest of it at Zinput. Although reflection can affect transients.
For receiving side the field incident on antenna should induce voltage and current again some voltage will be dropped at Ra2 and will cause reradiated power and rest of power will be dropped at Zinput of TL1 taking RL as load and ultimately dissipate at Rl.
I am not getting except of improving transient what is need of matching antenna to transmission line.