In Marketing Cloud, if I begin deleting data extensions, could that do anything to improve the overall loading speed of the UI in the various contexts where Contact Builder data is being referenced?
It would depend on how many you have vs how many you would be deleting, but in general, deleting Data Extensions can improve load time for the UI anywhere pagination is involved.
Deleting sendable Data Extensions can also improve Contacts Delete processing speeds if you're utilizing that feature.
If you're referring to Contact Data, as in the Data Extensions linked in Data Designer, deleting data extensions will not help load times unless you first unlink them from Data Designer.
In the above, note "Data Extensions in an Account" and "Objects in a Folder" as these would apply in this scenario.