The distinction you are thinking of is willing vs unwilling cuck. They are still both cuck MC stories. It's quite different from the brand of NTR we see in this game and many Japanese NTR with the "oblivious MC."
The unwilling, but knowing, cuckold you are talking about is pretty limiting in the ways that the story could progress. The focus shifts a lot towards the MC humiliation and cuckoldry instead of the female moral degeneration and cheating aspects. It's also pretty hard to write any justification for why an unwilling cuckold would stay with the status quo instead of blowing a fuse and going on a rampage. It can be done, for sure, but that also shifts the focus onto the male MC's powerlessness and predicament, which isn't the main thing fans of the oblivious MC + cheating slut NTR are looking for.
Edit: This post is talking about the meat of the NTR progression. Anything can go in the final stretches of the NTR whether it be willing cuckold, unwilling cuckold, oblivious NTR in perpetuity, broken MC, revenge MC, break up, fixed relationship, etc etc. It's all possible at the end.
