I once restored a Firth, Son & Co flute that looks very much like that one. Take a look at the pictures in the link below and compare details of keys, block work, etc. There are several very distinctive features that give me high confidence that that flute is of a similar provenance. For example, the round, flattened short F touch with a recess in the body, the style of the interlocking foot key touches, the matching style of all the keys and all the block work throughout. All of these are quite typical of American flutes of that genre. The rings are different, of course, but that is a fairly superficial detail and I have seen other flutes from this (and related) makers that have that style of rings.
The makers marks on these are often very faint, so it would not surprise me at all if the seller thinks it is unmarked, even if every piece is marked but only visible in really good light with a magnifying glass.
In the link below you can click on the images to get larger versions to see the details in close-up.
http://jonathanwalpole.com/firth-son-co-8-key
These can be very good players at modern pitch. $175 is a steal! I’d buy that in a heartbeat.
It should be easy to get a head made for it, but you may even be able to get one of your existing flute heads (with barrel) to work on it as is, with a suitable amount of thread wrapping on the tenon.
In this old thread, tstermitz describes a flute very similar to the one shown in your pictures. His even had the same wide bands on the rings. He links to a IFS sale of an Olwell restored Firth, Pond & Co flute with the same rings.
https://forums.chiffandfipple.com/t/firth-son-co-8-key-flute-cocus/104730/1
Oh, and it looks to me as if the original barrel is there in the first picture. It probably developed a crack which was fixed by an external nickel sleeve over the wooden part. If you look carefully, you can see the wood in the socket. If you get a head made it would be worth getting both a new head and barrel made, perhaps reusing the ring and band from the original. That way you can get a slightly longer head that doesn’t require quite so much tuning slide extension for A=440 hz playing. And it will be lighter than using the original, externally lined, barrel.