The OP is correct that origin of the phrase goes back to the time of the British Empire. The OED quotes exmples from as far back as 1746 and 1793. (See below.)
From the Oxford English Dictionary
man on the spot n. a local official, agent, or informant of a
government, company, news agency, etc., esp. in a foreign country; a
person with immediate responsibility or authority; (also) a local
The two earliest examples from the OED are:
1746 Laws, Ordin. & Instit. Admirality Great Brit. II. x. 98 If
there be no Consul, nor any other English Man on the Spot, in that
case the said Goods and Effects shall be committed to the Custody of
the Cadi of the said Place.
1793 Parl. Reg. 1781–96 XXXIV. 115 We shall have a man on the
spot, cloathed with the character of an Ambassador, that we might be
in a situation to treat with France.
Graham Greene used the phrase in his 1955 novel The Quiet American, which is about the waning days of French and British colonialism in Vietnam. Source, OED
I always like to know what the man on the spot has to say.
Finally (although there are many more examples), the phrase appears in the title of the book The Man on the Spot: Essays on British Empire History (Contributions in Comparative Colonial Studies) by Roger D. Long. A short summary of the book is given below.
Focusing on the role of the individual in the periphery of the Empire,
this volume illuminates John Galbraith's thesis that events on the
periphery of the British Empire led the man on the spot to expand
the area of British control. The man on the spot was a factor in
imperial expansion as much as, or sometimes more than, imperial or
company policy, which often opposed control of further territory
because of the expense.