It's not a 71, I think that's an industrial buttonholer or some kind of speciality purpose model. From the top picture I would say it's a 27. It could in theory be a 127 too, but easily noticed by the release loab on the top tensioner (127 doesn't have the loab, it has thread release in the presser bar lift). I wouldn't trust Singers serial number list completely, it has a few mishaps here and there. Generally it's trustworthy, but parts of it are missing for some factories and some numbers are more of a guesswork because of it. Yours look like it has been misplaced between the two models when they were made.
Your 27 looks fine, just a bit dirty. With full set of accessories and a straight spool pin is more than I could say of my 201 when I got it. There are signs of age and wear, but that's something I allow on my machines. I usually take a gentle approach, and do a bit now and then to improve on it. Replacing decals can be worth it in some cases, but it needs to be done very neatly. Lifted veneers on the cabinet can be glued back in place, some steam it flat and use a vice until the glue has dried. Some get very far with the right kind of furniture polish, others need to sand it down and refinish the top. I think I need to refinish the top of the cabinet on my 201. Your cabinet looks very much within acceptable, it needs cleaning and polishing.