I'm wondering if this baptism for 1820 could be James, that it was a late baptism and original birth year not recorded.
Jas. Brindley bp.20/6/1820 Burslem St John, son of Josh. (potter) and Elizabeth, abode Corbridge.
Various thinking's for this: there is only one marriage between a Joseph Brindley and an Elizabeth and that was 11/5/1788 Stoke on Trent to Elizabeth Sonkey. 5 children baptised Burslem 1791-1801. In 1841 a Joseph Brindley b.1766 Potter in Hanley, in 1851 your James states born Cobridge, in 1861 born Burslem, no other baptisms to a Joseph and Elizabeth 10 years either side of 1820.
Worth considering.
Annette