Am I being too nosey looking for this father 😚
No, it's reasonable to want to know.
The fact that he is listed on the birth certificate means that he did acknowledge the child by accompanying the mother to the Registrar's when she went to register the birth.
As the father's name is on the birth certificate, does it give his place of residence too? And did you notice whether or not there is an RCE (Register of Corrected Entries) listing attached to this certificate? It's unlikely but just possible, so do check.
Remember that a carpenter may also be described as joiner or (square-)wright in a census.
Normally one might turn to the Kirk Session minutes for further details. However (a) they are unlikely to tell you anything much that you don't know from the birth certificate (b) the KS minutes of Brechin Cathedral Kirk don't seem to be in the National Records of Scotland, though the Brechin East KS ones are and (c) by 1872, the kirks in many towns were beginning to lose track of errant parishioners as populations grew and people attended different denominations. So while there might just be something I doubt if it will help much.
I think you just have to plug away looking for a Robert Mitchell who was a carpenter or similar. I would go for the 1881 census, because the LDS Church transcribed the whole of that census, and it was made available on a set of easily searchable CD-ROMs. I have a set somewhere but they won't run on this PC (Windows 7). Maybe someone who has access to a set and an antiquated PC (Windows XP for example) would be able to take a look on your behalf. I might be able to do this in a couple of weeks when I can get my hands back on such a PC.