At this point there were still some darker spots but I was sure I had gotten the varnish off so thought it might be old stain that would come off.
Do I try another bleach round for evenness?
No I don't think the table is ready for that.
Although I'm not recommending other aspects of the video specifically (or the channel itself), take a look at the really consistent colouring of the wood surfaces revealed by the use of the sander in this video from Modern Makeovers, timestamps 2:49, 3:06 and 3:14. That's pretty much the level of uniformity one should be aiming for when stripping is completed1.
Now the wood in the video is something like maple or birch, both close-grained woods unlike the oak you're working with, but the same principle does apply up to a point — ideally the wood should look basically like new when you're done stripping and a modicum of scraping or sanding has been done.
With oak (also ash, chestnut, sassafras and elm) one might have to accept remaining finish left in the open grain because scraping or sanding to the bottom of it would simply remove too much wood. This goes double if there's any veneer present, as there appears to be in the central field of your tabletop.
Stripping in the real world
Stripping manufacturers don't want buyers to know this as it'll put a lot of them off, but an old standard for stripping by hand, i.e. not done by dipping (full immersion in tanks of a stripping agent) was that everything should be stripped three times....... Once is pretty much never enough. Two frequently leaves a little bit behind in some areas (which might or might not be removable by other means such as a scraping or a wipe with lacquer thinner or acetone). Three is sort of the sweet spot to getting the wood really pristine, at least with a really good stripper....2
1 Some stains that appear to be from other effects might still be present and require separate attention, e.g. oxalic acid for dark stains or solvent poultices for oil stains. Note that some stains can be stubborn and not respond to anything unfortunately.
2 It's important to mention that because non-pros in most parts of the world are no longer allowed access to strippers based on methylene chloride (because of hand-wringing worries about consumer safety and campaigning from pressure groups) the simple fact is that most consumer-level strippers are made from less-effective chemicals. It's widely documented in user reports how ineffective some of these are against certain finishes or layers of finishes, requiring multiple rounds to get the job done. As a result it wouldn't be impossible to take on a job that requires stripping five times or more if there's enough finish present, or it's particularly stubborn chemically :-(