Can private pilots run other non-passenger flight businesses?
According to the current FAA stance on the issue, none of it is allowed except glider towing.
The latest interpretation from the FAA that I'm aware of is the 2010 Perry interpretation, however, there was a conflicting letter (Del Rossi) from 1987 that said the opposite from the White interpretation.  The FAA further clarified the discrepancy between these two letters in 2010 with the Perry interpretation. (Note: this link contains both of those interpretations from 1987 and 2010 together.)
The FAA acknowledged the discrepancy in 2010 and determined that the older letter took precedence because it was issued by the national office, but the White interpretation was issued by a regional office, although on a later date.
For aerial photography and survey, if it is in furtherance of their own business and is simply incidental to that business, yes, they may perform the work as a private pilot according to the White interpretation.  The other interpretations, which take precedence say this is wrong.
If someone else hires them to fly so the passenger may do those things, it's definitely not allowed nor has it ever been mistakenly indicated as allowed by the FAA for private pilots in past interpretations.
There is the White interpretation from 1995 that incorrectly states:

To the extent that your operation would be in furtherance of your own
business of aerial photography or survey, and you are not carrying
persons or property for compensation or hire, then you may do so
holding a private pilot certificate. However, to the extent that your
operation would involve operating your aircraft as an aerial platform
for other photographers, etc., for compensation or hire, you would
need to be the holder of at least a commercial pilot certificate in
order to act as the pilot.

So, in short:

Do aerial photography/videography and charge for it? - No, but it was mistakenly allowed according to the White interpretation in 1995.

Do reconnaissance/similar and charge for it (aerial survey)? - No, but it was mistakenly allowed according to the White interpretation in 1995.

Deliver packages, or indeed cargo? - No, this would be carrying property for compensation or hire. See 61.113(a)

Drop skydivers - No, this would be carrying passengers for compensation or hire. See 61.113(a)

Pull up gliders for pay - Yes. See 61.113(G)


If you're actually planning on flying and making money somehow related to that flying and you aren't 100% sure you are legal, please ask an aviation lawyer in addition to online research.
If even the FAA managed to make a mistake on the subject, there's plenty of room for the rest of us to do so.