Same reason they're protective of their lawns

I wanna say it's maybe at least partly due to insecurity and a way for them to cope with their own inability to find and maintain relationships with women. If they have issues dating, is much easier to blame some nebulous "corruption of white women" than to admit and analyze their own failings. Maybe somewhat similar to blaming immigrants for "stealing our jobs".

Not protective of, controlling of.

Conservatives are very concerned over what they view as thieves stealing and contaminating their pure aryan breeding machines

They're misogynists and racists and that's the natural intersection of their two passions.

mostly it gives them a justification for doing violence to "The Other".

Like it's a flag they can wave and say "Look what the (slurs) did to our white women!". afaik it has nothing to do with the actual well being of any actual women, it's just an excuse for violence.

i don't think its restricted to whites. such misogynist views transcends race (eg Twitter 'discourse' around Zendaya and Tom Holland).

they view women as their property

Half of them don’t get laid, so they get mad that the “supply” is going down and getting “acquired” by other races

The other half piss themselves over the idea of purity being tainted

But this applies to any racist misogynist, not just whites

Chuds act like they own white women

There you have it. They view white women as their property.

Its an easy flag to rally around. Protect the women (because they can't see what they should be protecting themselves from in this world full of dangers), protect the children (because what monster would want to let kids be put into harm's way?), protect my property (its my stuff damnit, I paid for it, why should anybody else get to use it?).

A lifetime of chivalry instilled via pop culture

I think there are a lot of threads that come together that amount to capital's exploitation of women for unpaid social reproduction as a source of surplus value. I don't think chuds like explicitly have a list like this but they have internalized these concepts ideologically to the point that it doesn't need to be explicit (although if pushed sometimes they will admit it):

1. Bourgeois ideology is the primary ideology, and bourgeois symbols of wealth are aspirational goals for petty bourg and workers by default.
2. The forced exploitation of women requires buy-in from women. We know religious trad wives exist, I have some in my family. Some large percentage of women have to be made to fear society and hence desire to be "taken care of" (owned) by a man which is again bourgeois as the wives of the bourg are exploiters, not exploited.
3. Whiteness is the "highest value" race, when value is the product of an unconscious process of comparison
4. The bourgeoisie promises a woman to do domestic work in the home if a man will work hard enough to "provide" for her and their children.

So basically the obsession with white women is the false promise of a "high value" house servant/prostitute to the petty bourg, ideologically committed or naive, aspirational and reactionary workers, anti social and hyper exploited and backward men.

Like so many people have said, its about making women into property but more specifically turning women into capital that produces value for its owner, as well as carry out the crucial labor of social reproduction for free or at hyper-exploited wages