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The transitive and intransitive phrase to paint with a broad brush, and its variants, mean:
– to describe or portray someone or something in very general terms, avoiding or neglecting the finer details;
– to make a general or sweeping statement.

This phrase alludes to a style of painting characterised by the use of broad brushstrokes—as exemplified by the following passage from On the Exhibition of Paintings by Modern Artists, published in The Scots Magazine, and Edinburgh Literary Miscellany (Edinburgh, Midlothian, Scotland) of April 1822 [page 508, column 1]:

Mr. Thomson appears to paint always with a broad brush, and to apprehend that minute finishing of the details of a landscape may hurt the general effect, forgetting that it is in the combination of truth, and minuteness of finishing, with breadth and generality of effect, that the perfection of the art consists.

These are, in chronological order, the earliest occurrences that I have found of the phrase to paint with a broad brush and variants:

1-: From The History of Greece (London: T. Caddell and W. Davies, 1808), by the British historian William Mitford (1744-1827) [Vol. 4, chapter 32, page 147 (footnote)]:

Plutarch ¹, who commonly paints with a broad brush, regardless of nice distinctions, and often indulges in a very indiscriminate use of hard names and foul language, calls Nypsius’s troops altogether barbarians.

¹ Plutarch (c. 46 – c. 120) was a Greek biographer and historian.

2-: From a letter to the Editor, entitled Feelings of a Foreigner in America, published in The Newcastle Magazine (Newcastle-upon-Tyne, Northumberland, England) of March 1825 [page 109, column 2]:

I will […] endeavour to make myself familiar. This, I believe, I cannot better effect, than by laying before you my portrait, as drawn by the hand of my enemy some years ago […]. “Mr. Christopher Chatter,” says the arch rogue, “or Kit. Chat. as he is familiarly called […] is a long lean man, carrying about as much flesh on his bones as would feed a dog twice. [&c.]”
This is laying it on, I trust you will think, Mr Editor, with a broad brush and a hasty hand.

3-: From A Letter from Madrid, by ‘Vedette’, published in Blackwood’s Edinburgh Magazine (Edinburgh, Midlothian, Scotland) of June 1854 [page 684, column 1]:

What I have now written to you must be understood as applying especially and exclusively to Madrid, of which Ford gives a severe but true description, when he says, that “it is the centre of empeños, jobs, intrigues, titles, decorations, and plunder, to which flock the vulture tribe of place-hunters […].” The colours are here laid on with a broad brush, but they are not the less lifelike.

4-: From Substance of a lecture delivered at the Smithsonian Institution on a collection of the charts and maps of America, by J. G. Kohl, appended to the Annual Report of the Board of Regents of the Smithsonian Institution (Washington, D.C., 1857) [XIII.—On the arrangement of the collection, page 134]:

For many branches of natural science we possess no special maps of small territories at all; and for some, probably, we never shall possess them. Many natural features seem to sweep with a certain uniformity over a large tract of country; so that nobody has ever thought of giving us a special wind map of the State of Delaware or a zoological map of Long Island.
It is true that even in these extensive natural phenomena, which we now portray only with a broad brush, we may, in time, discover some regular local peculiarities worthy of being delineated on a map.

5-: From a review of The Cornhill Magazine (London, England) of August 1860, published in The Atlas. A Review of Politics, Literature, Art, Banking, and Finance (London, England) of Saturday 4^th August 1860 [page 627, column 2]:

Mr. Thackeray’s ² contributions are the second portion of his “Four Georges,” in which George II. is very neatly anatomised, and some of the peculiarities of the age painted in with a broad brush.

² William Makepeace Thackeray (1811-1863) was a British novelist.