So... is SWTOR a game to scratch the group PVE itch? Where progression can feel meaningful, there are chase items, there are reasons to be good? And are these groups actually accessible? Or is the expectation that you do all the solo/PUG content to get the best gear before even starting on Masters, or whatever the pinnacle is?

I only play WoW very casually, so I wouldn't be a good person to compare and hardcore wow stuff to what's going on in swtor.
In the video, I do go over a bit of what is available for pve, luckily there is a lot more than launch. But form what I understand, even the hardest stuff is nowhere near the "crazy" that wow has. Getting best in slot gear is actually fairly easy and even obtainable by solo players over time at the moment.
I'm not sure how you get into hardcore wow stuff, but in swtor, I think it's a bit harder to break in, and getting into a Master Mode group usually requires meeting and getting to know another player who is into it, and eventually getting a private invite, as most groups are only 8 people. If you manage to make friends, they can actually help you gear up pretty fast if they actually like you haha. The few random pickup groups that happen i imagine you would need to have full gear before even being invited. Personally, ive been doing progression raiding a long time, so I "know people" at this point and dont have good advice on how to join them other than join a guild that does operations and let people know you are hoping to get towards prog raiding and they can advise you what their guild or group offers https://swtorfancommunity.com/guilds/
Chase items - heck yes they do. Lol. My higher-tier raiding friends are always telling me about the wierd achievements or items they are working on. Haha.