Remove direct bitcoin calls from qt/bitcoin.cpp

Summary:
Partial backport of Core PR 10244 (71e0d90876efd11e2a4aeb8f3f806c5a1fd54b42)
https://github.com/bitcoin/bitcoin/pull/10244/commits/71e0d90876efd11e2a4aeb8f3f806c5a1fd54b42
Progress towards T417

Test Plan:

make check
bitcoin-qt # sanity checking

Reviewers: deadalnix, Fabien, #bitcoin_abc

Reviewed By: deadalnix, #bitcoin_abc

Differential Revision: https://reviews.bitcoinabc.org/D2813