If it is an adaptation of a surface grinder the handle at the top of the column should have a dial with graduations of .0001". The dial for the Y control handle should have graduations of .001" or .002". As Mark said this machine does not have the pulley ratio for grinding. Grinders of this era were fixed at 3,600 rpm + or- a few rpm.
Some of the smaller benchtop horizontal mills didn't have an over arm. There is currently a Burke #1 and a Goodell Pratt on eBay that don't have over arms. The Goodell Pratt table is at a fixed height with the head being able to be raised and lowered by the crank at the back of the machine. Given the age of the machine you are looking at I believe it is and always was a horizontal mill. I also googled "antique bench top horizontal milling machine" and found several images of similar size machines without an over arm.
Burke:
https://www.ebay.com/itm/Early-Burke-Horizontal-Milling-Machine-No-1/292670008749?hash=item44247dedad:g:I2MAAOSwk1NbaAx0&_pgn=1&_sacat=0&_nkw=horizontal+milling+machine&_from=R40&rt=nc&LH_TitleDesc=0|0
Goodell Pratt:
https://www.ebay.com/itm/Vintage-Goodell-Pratt-No-644-Miller-Horizontal-Bench-Top-Milling-Machine-refurb/202311081530?hash=item2f1aae1e3a:gX0AAOSwUQlavQYP:sc:FedExHomeDelivery!53711!US!-1&_pgn=2&_sacat=0&_nkw=horizontal+milling+machine&_from=R40&rt=nc&LH_TitleDesc=0|0
I don't believe the collets are 5C. They may be 2S or 3C, both of which I believe go up to 9/16". The 5C collets need a bore large enough to handle a collet that will accept 1 1/8" stock. Looking at the picture of the collet in hand I can't believe the body is big enough to accept 1 1/8" stock. Also note the arbor which is .750" where the cutters fit, but probably no larger than .500" where it enters the collet.