You could be right, but you would have to look further back probably.
I found these bits of info:
"In 1672 there were besides those mills belonging to the lord, three mills belonging to Henry Haden, who had inherited two from his father. It seems to have been the duty of the Lyttelton’s to repair the bridges at Malt Mill and Cornbow. In 1668 the bridges were in such bad repair that the constable prosecuted Sir Henry Littleton for refusing to store them."
"The abbot had the town of Hales created into a borough in 1270. The first dated evidence of such is in 1277. The abbot also had two mills worth 20s a year in 1291 and the new mill of Hales is mentioned in 1293. This may have been the Cornbow Mill. The Grange, to the south of the town, was probably the Great Grange recorded in the Manor Court Rolls in the 1270’s. These were collection points for agricultural products. A grant in 1344 to the abbot and convent of a weekly market on Mondays and a fair for four days at the Feast of St. Barnabas (11 June) may have superseded the previous market day and fair. But the Black Death arrived in 1348 and the town was decimated over 40% of the population succumbed to the disease."
"The site of the Cornbow Mill [HBSMR 4632] identified as a malthouse on the Plan of Halesowen of 1845 and as The Institute on the Ordnance Survey map produced in the 1880s. The malthouse, which was probably converted from the mill, was recorded in 1778 following the death of its owner William Powell. The site retains two sluices and three bricked up arches in addition to sandstone block and cobbled surfaces "
So, the Cornbow Mill may have been 13th century but the name still in use today.
Have fun looking into it, and good luck sorting out where everywhere was. Don't forget that Halesowen was in Shropshire until 1844