TL;DR warning I guess....
The first quarter-section of the table run we stained was left on too long (~8 minutes). It was clumpy when wiping off the wet stain and it dried looking streaky and uneven.
Note this does not apply here, I'm including it for completeness. For future reference the ideal time to tackle streaking in a stain job is right then, not later when the stain has had a chance to fully penetrate, and dry. If you notice some streaks or other irregularities re-wetting the surface (either with water, or as some guides suggest with more stain) it's possible to improve the result markedly with only a little extra effort. One additional point worth mentioning now is that when staining, especially with waterbased and spirit-based dyes or stains, it's absolutely vital to stain each surface in one go, so you need a method to get the stain on in bulk quickly and efficiently. This is why brush application of these stains is a bad idea (fine for oil-based), and you should for example use a rag wet with plenty of the finish so you can get the whole surface wet in one shot.
However from what we can see in the photos no tricks would have gotten you the smooth result you were going for because the wood wasn't prepped sufficiently for staining to commence. So let's back up a bit and talk about what should have been done before the stain went on.
When preparing wood for refinishing — not just when staining, but especially when you are — it's important to check for any areas of remaining finish (and possibly for glue contamination from the glue-up in the factory) before going to the next step. As I say in the Comments above, stubborn patches like this are very common (regardless of the method used to remove the old finish, including when stripping). Visual inspection in a raking light is one good way to spot these as you may be able to see marked differences in surface gloss, but the ideal method is to wet the wood.
And when planning to use a waterbased stain you want to pre-wet the wood anyway for other reasons1, so this is no longer an extra step.
When you wet a piece that's being refinished any area which still has finish or glue on it will stay light in colour while the bare wood will go significantly darker and you can immediately spot any problem areas. Even areas that only have a trace of finish or glue left in the grain or surface fibres show up and now is the time to find them and deal with them.
You really have only two choices here, actually three if you include repainting :-)
If you don't repaint2 and you want to stick with your original strategy you have no real choice but to take the tabletop back to bare wood and begin again as waterbased stains require a pristine bare-wood surface to work as intended. This is going to require sanding and lots of it, which is another argument in favour of not sanding old finish off in the first place since this is now two rounds of heavy sanding, increasing the chance of damage or alteration of appearance3.
The other option is to switch colouring products and go with a coloured overcoat, specifically "gel stain". This is not stain despite the name, it's thickened coloured varnish. "Gel stain" doesn't rely on penetrating the wood since it's a surface coating. But you would need to go a lot darker for it to possibly hide the current light patches so you may want to sand back all or part way for this too. However it's still worth considering as getting an even colour on a large surface is far easier with "gel stain" than it is with waterbased stains.
A note on pre-stain
Pre-stain treatments including commercial products such as "wood conditioner" and "blotch control" are useful, sometimes vital, to good results4. But anything like this should only be used where it's required and never otherwise. Wood species that don't require them (and this is the majority of hardwoods) will be partially sealed, greatly limiting how much stain they can absorb and therefore how dark the wood can go.
If you'll be doing more refinishing jobs I strongly recommend you switch methods for removing the previous finish and primarily rely on chemical strippers instead of sanding, and where they can't be used doing as much scraping as you can. Some sanding is inevitable and necessary, but as covered in numerous previous Answers of mine sanding is the worst way to remove old finish and it should never be thought of as the primary means to do it.
1 You're raising the grain ahead of time so that the stain doesn't do this, but in fact you may actually want to dampen the wood slightly just before the stain goes on (in addition to these previous wettings) to ensure a good result.
2 And jokes aside I would suggest you give the idea some consideration as it would require far less time, effort and cost — you can go straight to this option after a light sanding.
3 Edges and corners are susceptible to damage when sanding even by hand, but much more so when using power sanders. And any heavy sanding risks ruining the appearance of edge treatments like chamfers and roundovers. There's even the chance of visibly thinning a panel that's not particularly thick to begin with.
4 Note however that most are nothing more than diluted finish, which you can of course make yourself at home and at a significant saving. In the past, before such products were available, 'size' (watered-down glue) and heavily thinned shellac 'washcoats' were used when needed and can still be relied on for this purpose.