I started doing the "walking around" bit after negative feedback ~20 years ago, and stopped again pretty soon after it was clear that while it works, it's a band-aid for getting people confident that it's ok to talk to you whenever, and the real problem was that I'd been pretty burnt out and shut myself off for a while, not that I didn't literally walk the room.
With remote teams what I've done as a workaround with some teams have been to enforce a short 1-on-1 weekly, and tell people if we don't have anything to talk about, then we make it brief. It prevents team members who are reticent about taking your time from having issues build up because they feel it's too small or too personal to schedule a meeting about when they have a time slot they know you'll schedule regardless. For some it was almost always a 5 minute annoyance of me asking if there were any issues, them saying they had nothing to talk about, and for us to leave it at that. But it acted as a safety valve, and every now and again something absolutely critical would get brought up in those calls.
And many times when they had "nothing" to talk about, I'd ask them followup questions about small stuff like e.g. how they got on with specific people, or how a certain project was going, and suddenly it turned out they had plenty of things to talk about.
In teams where you know everyone will speak up, you can dial back both the actual and "virtual" walking of the room, knowing they'll come and tell you, but "walking the room" effectively acts as a way of lowering the barrier to bring up an issue, because they're not "bothering you".
With remote teams what I've done as a workaround with some teams have been to enforce a short 1-on-1 weekly, and tell people if we don't have anything to talk about, then we make it brief. It prevents team members who are reticent about taking your time from having issues build up because they feel it's too small or too personal to schedule a meeting about when they have a time slot they know you'll schedule regardless. For some it was almost always a 5 minute annoyance of me asking if there were any issues, them saying they had nothing to talk about, and for us to leave it at that. But it acted as a safety valve, and every now and again something absolutely critical would get brought up in those calls.
And many times when they had "nothing" to talk about, I'd ask them followup questions about small stuff like e.g. how they got on with specific people, or how a certain project was going, and suddenly it turned out they had plenty of things to talk about.
In teams where you know everyone will speak up, you can dial back both the actual and "virtual" walking of the room, knowing they'll come and tell you, but "walking the room" effectively acts as a way of lowering the barrier to bring up an issue, because they're not "bothering you".