Glycolic acid (hydroxyacetic, HOCH2CO2H) is a mild organic acid, not very expensive, and the calcium salt is quite soluble. It is essentially a non-smelly vinegar! It has been used for brick washing.
One issue is that whereas HCl will dissolve CaCO3 until the HCl is all gone, the pH of glycolic acid solution increases as it dissolves CaCO3, so the dissolution reaction slows down and finally stops at maybe 50% completion.
Citric acid and tartaric acid are very soluble, but their calcium salts are not. Calcium citrate and tartrate may come down as fine crystals in the pores of the brick and be difficult to remove. Calcium formate is moderately soluble and so would be useful, but formic acid is a vesicant, so it must be used with as much or more care than HCl.
Muriatic acid is readily available at 32% and 20% concentrations. Diluting by a factor of 50 would give a pH about 2, which is safer for people and brick, but that's a lot of water to clean a brick wall.