It probably just doesn't have it.

If you hunt around in your \windows directory, you could find the folder with cursors in there. I'm not sure how the structure is set up, as I never installed the 'extra' cursors, but you could copy a 'crosshair' cursor from another set in there so you don't lose the cursor.

Also, have you tried looking at it on a black background? Maybe it's just completely white, and gets lost in the page. I know the standard one sometimes gets lost in a gray background.

I avoid websites that change my cursor like the plague.

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Well, considering how 'changing the cursor' is as simple as using cursor: whatever; in CSS, and that this technique is often used to give hints to an action on a webpage. For example, links turn into a hand when you mouse over them, text input turns into a text selection cursor, etc.

Sometimes, an element is formed in the way that the function is non-obvious, and so mouse cursors are used as hints. Of course, in good design, it appears almost transparent to the user, as the behavior is familiar.

It really only becomes noticeable when the cursors stick out due to poor design choices, which may very well be the sites you want to avoid.

Note that this is functionality built into most css-aware browsers, and not anything nefarious like Comet Cursor or a variety of other undesirable programs.