Arguable - I'd say even unlikely in most real world situations. The US has recently had a wildly unpopular president who showed remarkable indifference to said public opposition. And relying on the media to gauge public support, or inform the populace in objective terms about the actions of their representatives, is inadvisable. It is not difficult to find instances of the same media organizations arguing both in favor and against public policies depending on the elected official enacting them[0][1]. In that regard, mainstream media seems more akin to a tool of propaganda, shaping opinion and manufacturing consent after the fact, than a useful organ of democratic governance.

The issue is that the control the population realistically has over their representatives in a democratic republic like every (?) modern democracy is by necessity of the size of said governments extremely limited. You can't have a referendum on every issue, so for the most part all the public gets to decide is if someone did a good job over the last X years. I recommend reading Democracy for Realists for a more in-depth look, but suffice to say that that level of interaction is far too coarse to guide public policy on immediate, subtle, and complex issues like Ukraine, COVID, or Chinese foreign policy. If something is not a core issue (i.e. something that the parties have split and staked their identities on, like women's rights, access to guns, immigration, etc) or is so universally desirable that everyone just wants more of it (i.e. visible short-term economic growth - note that long-term economic growth is generally not rewarded by the electorate), policymakers have essentially no incentive to actually care, because their voters won't care enough about those issues when it matters to sway their vote one way or another.

To your last question; the public's opinion of a decision doesn't influence whether a policymaker will make the decision. The point of the media is not to inform the public on what their representatives are doing, it is to shape the narrative of politics in the country. The actual decisions, the policy positions, are for the most part incidental to that. The causal arrow goes the other way. See also: [2][3][4]

[0]: https://mobile.twitter.com/BaldingsWorld/status/153358567932...

[1]: https://www.racket.news/p/the-sovietization-of-the-american (search "Khashoggi")

[2]: https://www.motherjones.com/kevin-drum/2019/01/marc-edwards-...

[3]: https://www.reddit.com/r/chomsky/comments/c1fk5l/newsweek_in...

[4]: https://en.wikipedia.org/wiki/Propaganda_model