Wait til you hear about these guys...
A lot of the GOP influence comes because of geography. If you look at who voted blue or red in recentl electsions, it's urban vs rural. Also the West and South are very GOP overall.
Because many LDS live in the mountain west, they tend to go with the politics of their geographical location.
Also our culture is conservative overall. Plus some of our doctrines lead LDS to feel they must vote GOP.
And, yes, as you have said, we had some strongly conservative leaders in the 20th century. However, I was happy to see that as soon as President Benson became prophet he very much toned down is personal politics, to put his focus on the gospel. One of the best talks ever is his Beware of Pride - I review it regularly, and would vote to canonize it in some form: Beware of Pride, https://www.churchofjesuschrist.org/study/ensign/1989/05/beware-of-pride?lang=eng
The history behind the current GOP is largely described in the following book:
Born Fighting: How the Scots-Irish Shaped America, by Jim Webb, https://www.amazon.com/Born-Fighting-Scots-Irish-Shaped-America-ebook/dp/B000FCKGTS/
This Scots-Irish immigrant culture from 200 years ago has come to dominate the non-urban US.