And yet, Republicans (let's not argue who's a true conservative) have continued to put a lot of effort into fighting against LGBTQ protections in recent years.

On the subject of "enabling the suppression of their fellow man", the only really hot-button issue for Republicans that can truly be framed as them defending against their own rights being suppressed is gun control. The rest of their major issues either fall on the wrong side of suppression, or aren't really about somebody being suppressed in any direct way.

In fairness, there are other issues framed this way. The "overreaction" to COVID-19 is seen as suppressing economic rights and livelihoods.

Good point. Though a lot of the backlash against public health emergency powers actually being used when necessary is of a fairly knee-jerk, borderline anarchist nature (reminiscent of some recent anti-taxation movements that have been similarly vague). There's a difference between calling for government to have stronger protections on a particular right or freedom, vs simply rejecting any instance of government having a direct impact on your life.

It tends to be accompanied with a belief that things are not that serious or that there won't be a real benefit from the measures.

And the worst versions impute sinister motives among those implementing controls: that it's an attempt to suppress economic activity and make people slaves of the government/elites/etc.

For something you seem to think is "pretty rare"... I know quite a few people who would gladly discriminate against LGBTQ.

Indeed, a May 2020 Gallup poll says that 32% believe that same-sex consensual relations are wrong and 24% of the US believe that same-sex consensual relations should be illegal. So not only do they want to be able to not hire, they want people who like the same sex to face legal consequences.