As far as I can tell the only mention of "Eastgate Space" anywhere is on that sign. The Martins Bank building was finished in 1938 and
seems to have been remodelled a couple of times. The site next to it was used as a car park until National Deposit House was built (the car park underneath it still has the entrance in the same place). The building on the other side (7 -25 Eastgate) was originally called Shell BP House and
it looks like the gap was kind of a forecourt to the entrance, with (different) steps up from Lady Lane and a couple of trees.
If you're interested in council byelaws, there's actually a
Leeds City Council Act that was passed in 2013 to regulate the size of those pedlars' handcarts you get on Briggate.