...I wonder if in this same situation, wherein I am at someone else's home, they could have the FG set up and be the server-side host? Or is it that the GM always should be the server-host? (I guess I could go look it up in the documentation...)
The most important thing is that your campaign folder is there in the Fantasy Grounds data folder. FG campaign data is never stored remotely - always in the local data folder. I do FG server hosting at IRL tables, mostly to host players that can't make the session and need to connect remotely. I add a pseudo-player to the party and move their token around the map on an HDTV, so players at the table can have a player's view of the remote player(s) on any maps. And yes, I usually put that pseudo-player's combat tracker up on the big screen to show initiative order - many rulesets display info in the GM's Combat Tracker that's shouldn't be seen by players.
The other important thing is that the host computer have the necessary ruleset, DLC (adventures, companions, purchased maps, etc.), and extensions that are being used. If another player hosts, but doesn't own the ruleset and DLC it won't be available, as FG content is tied to a license. If I'm playing around a table at a player's home, I just bring my laptop along and connect it to their HDTV - all the necessary content is there. For one of our campaigns, 2 of my players own the campaign content and have nice gaming rigs and gaming tables. So I've used their PCs and FG licenses to host the session. In those situations, I just bring my campaign folder and any extensions on a thumb drive and copy it to their FG data folder.
Our general rule is not to have players at the table connect to the FG server via their laptop, but that's mostly due to player's preferring to have a break from tech. But if we wanted to allow that we could certainly accommodate it. If there's no remote players, you can even do that completely disconnected from the Internet, as FG has a LAN mode.
The tying of content to a license makes using FG a bit more challenging than roll20 or Foundry in such scenarios. There's no sharing of content between different GMs, so if you want have a friend do the hosting on their PC, they also need to own the necessary content. TBH, it hasn't mattered much to me, because I'm not big on reading content like GM guides, companions or supplements in-game and prefer PDFs on my tablet or another screen. And I now have the option of the Oline Reader too.

So when using those 2 player's gaming rigs, they've really only need to own the core ruleset.