Those are specific forms of reaching out. Contact is different but not any more efficient IMO, so I guess we will have to agree to disagree. If I had to direct someone to look into it, I might say --
'Please find out if Bob's surf shop carries Mr. Zog's Sex Wax.'
A few of the terms in the header image on the article seem perfectly appropriate to me depending on context, but I'm probably thinking of them being used literally (reaching out as in contacting, incentivising as in more money paid, deep dive as in RCA) and have insufficient exposure to the BS versions to comment - so, I'll stop!
The reason reach out is used is to give the individual being asked some autonomy / responsibility in the action.
Eg if you ask someone to call bobs surf shop and they don’t answer they might stop there or call back later. If you ask them to reach out they might go down there and see them, set up a meeting, speak to a friend that works there, etc.
Both Wiktionary and dictionary.com list definitions of "contact" as a verb. I don't have access to the OED at the moment, but I assume they would have it listed as a valid verb as well.
'Can you call Bob's surf shop and see if they carry Mr. Zog's Sex Wax?'