Horatio
You're welcome
They were worried about the fee rising and I also believe the club thought Valverde needed more options to cope with the heavy workload in the second half of the season.
Sure Coutinho can't play in the CL but adding him along with Dembele coming back from injury is a massive boost to Valverde and the team. It allows Valverde to rest the likes of Iniesta and Messi etc more often especially Iniesta who's 33 years old and doesn't have the pace nor energy like he once had to play on a consistent basis.
While there's players on this team that can afford to play a lot without to much issues, there's others though you need to be more careful with due to age and injury concerns.
Still there has to be more at play here than just tactical advantages. La Liga is as good as in the bag. He can't play CL and he sure as hell hasn't been brought in for Copa.
Like you said I think it's mostly because of the fee rising. Quite possibly Coutinho became really unhappy at Liverpool as well. Rarely have seen a player wanting to leave this fast sacrificing a round of CL.