Is the game itself stuttering, like framerate? Or is the framerate fine but things just pause on occasion and then the game feels like it's catching back up?
You make it sound like #2, which nobody's addressing here. It sounds like you're getting packet loss or packet delay somehow. (It should be noted that you can experience this with or without anything else running alongside FFXIV - really depends on your ISP and the route between you and the servers)
Do a ping test of your datacenter and that should give you a teensy bit more insight - I am unsure what all of the domain names are for each DC, but I know that NA's should be neolobby02.ffxiv.com - if you ping that and see high ping values or timed-out messages, that is your problem.
You can also download a portable program called Ping Tracer which can visualize the latency between you and the server.
Apex legends is an EA title, so do a ping/trace to easo.ea.com
individual spikes on individual routers along the path are expected as responding to ping requests is not high on a router priority list.
Also, you can check the status of the servers here: https://apexlegendsstatus.com/
I personally like to use Ping Tracer for keeping an eye on my connection. I haven't used WinMTR, but from looking at it, Ping Tracer is potentially an alternative and it can give you a visual representation of your connection.
Thanks. I've tried measuring latency and packet loss along the path to the east coast servers - 209.95.51.95 and 104.153.85.28 are the two I've found from launch.log, though I just noticed an IP in a different block, 74.201.228.184. Haven't found any issues yet.
I've been using pingtracer, https://github.com/bp2008/pingtracer, which pings every host along the route and shows a graph for each host. That can help you narrow down where the issue is - in your ISP's network, at the edge, in someone else's network?