In front of the banner advertising the RA Summer Exhibition, the swagger statue of Sir Joshua Reynolds (1723-92) by Alfred Drury stands garlanded with flowers. But the Academy he founded won’t be marking his tercentenary with a retrospective, just a small display and a series of artists’ lectures. For an anniversary show, you have to travel to his native Devon.

Ever since the Pre-Raphaelites dubbed him ‘Sir Sloshua’, Joshua Reynolds has been out of fashion: blame the outmoded ideals of beauty he promoted in his Discourses and his role as portraitist to the Georgian establishment. But Reframing Reynolds at the Box in Plymouth gives us a glimpse of a different Reynolds: one who didn’t always practise what he preached and could be tender, playful, even mischievous.

Intellectually curious, intensely gregarious and unusually well educated for an artist – his clergyman father was a master at Plympton Free Grammar School – Reynolds had a genius for networking. Within a few years of setting up his London practice in 1753 he was mates with Samuel Johnson, Edmund Burke and Oliver Goldsmith. His career was stellar. He rose to become the first president of the Royal Academy in 1768, a knight of the realm in 1769 and Painter in Ordinary to King George III in 1784, despite the king much preferring Gainsborough and referring to Reynolds as ‘poison in my sight’. He was a social climber, not a snob. He never completely laundered his Devon burr and refused to behave like a tradesman with titled clients. Some took offence (‘I never saw so vulgar & so familiar a forward fellow,’ huffed one aristo); others were disarmed. A sitting at his Leicester Square studio, where he kept a macaw in the house and a mangy eagle in the yard, was fun.

Since returning from a study trip to Italy in the early 1750s, Reynolds had stepped into the shoes of Grand Tour portraitist Pompeo Batoni as image creator-in-chief to the British aristocracy: his sitting room, recorded his pupil and biographer James Northcote, was crowded with ‘women who wished to be transmitted as angels and men who wanted to appear as heroes as well as philosophers’. His double portrait of ‘Mr Huddesford and Mr Bampfylde’ (1778) paints a charming picture of the tender feelings the new men of the Enlightenment were encouraged to cultivate. It ended badly for Bampfylde, who fell madly in love with Reynolds’sniece Mary Palmer, couldn’t cope with rejection, was arrested for smashing a window of Reynolds’s house and was eventually confined to a mental asylum in Sloane Street.

The humanity of this portrait and of Reynolds’s pictures of children – of which there is only one, a studio copy of ‘Age of Innocence’ (c. 1788), in the show – makes it hard to take his more pompous paintings seriously. What’s to be made, for instance, of the ridiculous full-length portrait of the Anglo-Irish aristocrat Lord Peter Ludlow (1755) in the uniform of a heroic Hungarian Hussar? That Reynolds kept a dressing-up box we know from the fact that three male sitters in this show are sporting the same natty red velvet jacket with fur trim. The Italian engraver Francesco Bartolozzi, a famous lady killer, looks particularly moody in it. (Gainsborough later questioned why this ageing lothario should ‘spend his last precious moments in f—g a young Woman, instead of out doing all the World with a Graver; when perhaps all the World can out do Him at the former Work?’ Miaow.)

The suspicion, with Reynolds’s more extravagant aristocratic portraits, is that artist and sitter are playacting. Yes, the sumptuous full-length ‘Portrait of Lady Elizabeth Keppel as a Bridesmaid’ (1761) performs the social function of advertising her marriageability, but depicting her – with an ‘exotic’ Black model in attendance – in the act of decorating a Roman statue of Hymen is an elaborate game of aristocratic charades played by a brilliant pasticheur of the ‘Grand Style’. Reynolds was a master of publicity. When his own name was announced at a social gathering, noted one of his servants, everyone turned to stare and cleared a path, ‘whereas when a duke or earl was announced, few of the people took any notice’.

A gambler and a drinker – he explained to Johnson that, in a long day, wine ‘puts me in the same state as when I woke up’ – Reynolds worked as hard as he played. ‘Shaved and powdered by nine in the morning, and at his canvass’, according to the Times, he was in his studio seven days a week from September to June. In a 50-year career – before cirrhosis of the liver carried him off aged 68 in 1792 – he painted more than 2,000 portraits, 661 of which are in British public collections.

The 30 paintings in this exhibition are just a taster of what’s out there. Now that his portrait of Omai has put him back in the news, could we please have a bigger exhibition?