Now that I've finished sanding, there are quite a few spots that have white splotches from the wood filler
That looks like something you get almost always when sanding wood that isn't absolutely perfectly flat — fine sanding dust settled into some minor surface imperfections (they can be incredibly small or shallow and still show this effect).
To check if this is the case give the area pictured a quick wipe over with alcohol or mineral spirits, if the whitish bits vanish they're sanding residue.
Brush the surface well and then wipe down with a dry or barely-damp cloth (microfibre cloths are great for this) and the light residue should diminish greatly, possibly disappear entirely1. See this previous Question for more if needed, How to remove dust after sanding before applying paint/oil/glue?
Also, secondary question: what will happen if I don't use a wood conditioner before staining?
There are multiple previous Q&As that touch on the use of "wood conditioner", IF you use it to help prevent blotching where and how to use it properly, alternatives to the commercial product (all cheaper, often more effective) and alternative approaches to colouring blotch-prone woods that sidestep the need to pre-treat them. I'll link to a few of the most relevant at bottom.
It looks like you're working on a lovely bit of wood, with nice tight grain not commonly seen these days on pine and related softwoods, so I would urge caution re. your plan to stain if you'll be using a true stain (penetrating stain). Pine is notorious for how badly it stains and that's why there are products sold to help get better results, but they typically don't completely solve the problem2.
The makers of "wood conditioner" (AKA "pre-stain conditioner" or just "pre-stain") can give buyers the impression that their products are a magic bullet for blotching but that's far from the truth3.
Some previous Questions:
Should you always use Pre-Stain?
Wood conditioner between stain layers
Should pre stain be used with gel stain?
Regardless of the finishing approach you end up settling on I would recommend you test it out on the underside of the slab before starting on the top (I'm presuming you don't have an offcut but if you do use that).
In either case it's important to sand the test surface to exactly the same level as the top.
Testing before committing to the completed item is considered a must-do in finishing as you can never be sure how an individual piece of wood is going to take finish, finishing products can go bad during storage or the test shows that you haven't stirred/shaken enough yet. And occasionally there can be some variation from batch to batch in colour, consistency or drying time, any one of which it's important to know about before the finish touches the workpiece.
1 Don't sweat this too much, it's not necessary to get every last spec of sanding residue from the surface of wood as the first coat of any finish will take care of the remainder.
2 The issues are that they A) don't reliably or completely prevent blotching and B) because they act by partially sealing the wood surface they limit the absorption of conventional stains, reducing the staining effect. Meaning you often get a lighter colour than you wanted. And on top of this, as Bob Flexner has highlighted in recent years, the instructions that come with many/most of these products are wrong about how to use them most effectively!
3 Regardless of whether you use a commercial product or a homemade solution (which as I say above can often work better) the problem is not always fully conquerable, and because there's no real way to preview the result some woodworkers take the position that they don't stain blotch-prone woods at all. They either exclusively colour them using coloured topcoats, or, only use these woods for their natural colour (bearing in mind that they will naturally darken over time from light exposure, sometimes considerably).