Would hard drive whirring at inexplicable times really be the result of the cache,
A modern PC has several caches in various subsystems, so simply mentioning "cache" without further qualifying what you mean (e.g. disk drive, processor, or OS) is ambiguous.
More often the OS cache will have data that was written out to disk or read-ahead of already open files, rather than guessing what "new files in advance" might be read.
or is it something else entirely?
Browsing the web can cause can cause disk access for loading fonts, but most likely the disk activity is related to recording the browsing history, authenticating and updating cookies & certificates, and writing a copy of downloaded web content (e.g. Firefox maintains a directory named "cache", and Internet Explorer also has a folder for saving "webpages, images and media" content named "Temporary Internet Files").
Web browsing today is a lot more I/O and computational intensive than back in the day when WinXP was introduced. The minimum system RAM requirement for WinXP was originally only 64MB. Back then you didn't have an anti-virus or anti-malware program running concurrently to protect your system. There wasn't animated graphics (e.g. Adobe Flash) or Java scripts or other add-on & plug-in programs running in frames within the browser window. All of these additional programs are executing concurrently with the modern web browser, and can cause additional disk activity.