Oh dear (again)!!!
Yup, from the pictures it's definitely a scroll saw - and just to confirm that, just as YOU say, the label says so too ;-)
You're definitely correct, it should definitely cut something with a new blade - including your finger!
You say that the motor runs and that the counter-balance weight goes round and round (as it should of course), AND it also sounds like the blade tension is correct, or maybe, even a bit too much.
So now the only thing I'm now wondering is re the hole in the face of the counter-balance weight - where the "crank pin" (as on a car engine for example) sits into the hole for it in the face - has it perhaps worn so much that with no load the arms go up and down OK, but as soon as you put any load on the blade (which is of course transferred back to the arms and then to the "crank pin" siting in the hole on the counter balance weight) does the drive to the blade become virtually nil because the hole has worn too large (or come to that, has the "crank pin" worn away to an oval shape)???
Sorry I haven't explained the above very well but if you have trouble I'll do a sketch and scan to post it here for you.
The only other thing that I can think of is the "gurt big spring" at the back of the upper and lower arms broken - or VERY weak?? I don't know your machine, but you may need to remove the upper outer casing to see that spring.
Anyway, good luck, from the pics it looks to me just like so many other "badge engineered" scroll saws that come from the same (?) Chinese factory - though I AM surprised at the blade length on your machine - all others I've seen of that generic type definitely take 130mm blades.
Perhaps someone else with more knowledge of your particular machine will be along soon.
HTH