There are plenty of pour-over coffee funnels with a flared bottom so the funnel can sit on top of a normal pitcher, carafe or coffee cup.
So presumably the pitcher doesn't need to cover the sides of the funnel. Try using a mason jar or any similar heat-resistant container that the funnel fits into the mouth of. If you don't notice any flavor difference, then you know you don't need the glass sides.
There are also pour-over sets that use a stand to hold the funnel, for example:
You could probably improvise a stand. All it really needs is a ring for the funnel to sit in, and a stand to hold the ring the proper height over your pitcher or carafe. An erlenmeyer flask stand or a tripod for using over a bunsen burner would work, eg
(This one is only 5" tall which might be a bit short, but taller ones are available.)
As moscafj pointed out, some carafes are made of boro-silicate glass so they can be used on the stovetop. If that is your intended use, be sure that your substitute carafe is stovetop-safe.