How?

I think we should bring back a version of fairness doctrine [1] or right of reply [2] targeted at platforms.

There are other ways of pressuring them at different levels as well: demanding moderation transparency, regulating them as we do for utilities, supporting competitive alternatives which respect free expression, etc.

[1]: https://en.wikipedia.org/wiki/FCC_fairness_doctrine

[2]: https://en.wikipedia.org/wiki/Right_of_reply

Liberals have dreamed of restoring the fairness doctrine ever since Reagan administration killed it for public airwaves and ushered in the era of right wing monopoly of talk radio.

I know that it was killed by Reagan's FCC, but I really don't have a good understanding of their motivation.

The Wikipedia article quotes someone in the Kennedy administration as saying: "Our massive strategy [in the early 1960s] was to use the Fairness Doctrine to challenge and harass right-wing broadcasters and hope that the challenges would be so costly to them that they would be inhibited and decide it was too expensive to continue."

So it kinda seems that weaponization of complaints was part of the issue.

Do you happen to have any pointers or comments on Reagan administrations reasoning, and the efforts by liberals to restore the doctrine?

You can read all about it in that wikipedia article. The long list of liberal efforts is in the section labeled "Reinstatement considered - Support"