Hi
My 4th gt grandmother was possibly the publican of the Coach and Horses. I've copied this from my search:
The Coach & Horses (no. 17), landlord in 1778 William Ball, in 1809 Daniel Gibson, in 1818-20 G Williams, in 1822-3 J Williams, in 1828 Joseph Sadler, in 1840 John Jones, in 1850 Elizabeth Jones, in 1871 George Edwards, in 1880 Robert Webb, in 1895 T Richards, in 1902 Mrs H Webb, in 1910 Mrs M Moore, in 1920 John Patten. The licence was withdrawn in 1922
Hope it helps. The pub is no longer there. I can think of only 4 or 5 pubs left in Handbridge nowadays. The actual Flint Mountain is at least 20 miles from Chester, in Wales so I'm not sure of that connection. I have not heard of any area still known as that around Chester. However if the C&H is where I think it was there was an area called the Rock nearby. It's a sheer guess but maybe it was jokingly called Flint mountain by locals.
Regards
Janice