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.