Where did the tendency for President Trump and Republican governors to downplay the need for COVID-19 safety measures originate?

It's difficult to answer this objectively, because there is a lot of disparity between the public stance of people and their actions. A lot of decisions are made for appearances sake, not on a rational basis.

The republican party, as a general rule, favors businesses over people and assumes that businesses will take care of people by themselves, through market forces and a trickle down economy. Happy businesses leads to happy people. Whether that works or not is not in the scope of this question.

Locking down businesses, requiring masks and hygiene standards as well as enforcing social distancing is bad for businesses, because they have less customers and there are additional costs for disinfectants, cleaning staff, home-office infrastructure and such. That would mean that businesses would have to go bankrupt and could no longer pay wages, unless they get "bailed out" by the government. Bailing the companies out costs a lot of money, which would increase the deficit. Without the bailout, the mass-unemployment would lead to an increase in suicides outweighing the deaths from the pandemic.

On the other hand, without locking down and with no mask requirements and such, the businesses can decide on their own rules, stay open, keep their customers, pay wages and not cost the government any money. The market forces will lead to customers and employees staying safe by themselves, because in theory customers will only buy in stores where they are safe and employees will only choose to work for companies that keep them safe.

In addition to that, an argument brought often in the early stages of the pandemic, it affects old, "unproductive" people more than the young and productive people, so having retired people die so that businesses can keep young people employed is seen as a preferable trade-off.

Whether those arguments are right or wrong is also out of the scope of this answer.