You are describing a property which has had little updating (which reflects obsolescence, a form of depreciation). An appraiser definitely should take the items you have mentioned into account. The "cost to cure" (cost to update these specific obsolescent items) should be reflected in an appraisal. Most agents will confirm that the appearance of kitchen and main bathroom, in particular, are heavily emphasizeed when marketing any house. And an "original" furnace, 40 years of age, is well past its "best before" date!
A professional appraiser does not have any "prospective interest" in the property, and should provide a written statement to that effect. A major point this would indicate is that he / she would not be intending to list the property for sale and therefore has no vested interest. A report produced by an agent, who does not have professional appraisal qualifications, might be questioned on that basis. (Full disclosure: I am an appraiser, and do not handle sales.)
I assume from your question that you are discussing a single residence, not a commercial or industrial property, which certainly can be appraised but would involve numerous other concerns.
If you are intending to take the appraisal to a bank for financing purposes, start by asking the bank for two or three names, acceptable to the bank.
Best wishes.