By closing up we defend us from some threats, but open gates wide for others. Foreign actors compete against much stronger domestic media machines and as you mentioned have to operate in foreign cultural environments. Gaining true influence also always involves financial flows, not just propaganda campaigns, so it is sure possible to mitigate these threats without closing information flow.
Consider the opposite threat of democracies being undermined from within. If some internal "threat actor" gets control of the executive branch and of the media and also can prevent information flow from the outside, very little can be done against it.
I think it is critical to keep in mind this second possibility even when the first threat seems more urgent.
Propaganda is not necessarily to gain influence or money. Eg: Country x just wants to mess with people's heads and turn them on each other to weaken a rival country. Or: Country y runs a crafted propaganda campaign against a rival. As a result, some sector of its own economy starts doing better at the expense of its rival.
>If some internal "threat actor" gets control of the executive branch and of the media and also can prevent information flow from the outside, very little can be done against it.
I understand the scenario (it's far from new), but that's what the design of any particular democracy is supposed to minimise. Term limits, separation of government powers, etc.
Step 1 is reduce your attack surface :) As a second point, democracies are propaganda campaigns - it's a feature, not a bug.
I believe that national cultural and societal norms play a key part in self-regulation. I think it's too much to ask for those balancing forces to work as effectively without first turning down the firehose.