The information I gave was from Kellys directories. It obviously referred to another British Lion. Directories often did not list the name of a pub, particularly beerhouses (which only had a licence to sell beer), but just referred to them as beer retailer (which could also include off licences).. Looking at the names of your ancestors, Joseph Armstrong in 1910 -1921 is listed as being a beer retailer at 96 Alma St (which is on the corner of Inkerman st), but has he gone by 1932. There is no Joseph Armstrong in 1908 listed , but in 1904 there is Armstrong Joseph, cycle maker, 348 Summer lane, Though he may be no connection. The electoral roll for 1930 lists 96 Alma St as The Dewdrop Inn
There is no listing for a separate Joseph Edward Armstrong in Birmingham between 1904 and 1932 (for years I have mentioned ). It does look as if the other Joseph Armstrong may, as I suggested have been the manager, but not the licensee.
Mike