The punch body itself is probably OK, although the retaining screw in the upper part is a weakness. Can't help that I am afraid. Just don't strip the thread. (It probably will strip eventually.)
Your problem is how the inserts are made. Typically they are made for mild steel and thick aluminium, with the necessary clearance between the male (punch) and female (die) bits. These clearances are necessary to allow the punch and die to self-align during use. However, the clearances are too big for really thin foil. Foil (SS, Al, Ti, brass, etc) will drag into the clearance and pucker and ... well, make a mess.
You can sometimes punch aluminium foil by laminating it between two sheets of similar aluminium sheet, but you need to clamp them together hard. You could try double-sided tape for this as well. I have done this with SS, Al & brass.
In theory you could also try this with Ti foil, but the usual Ti alloy (eg 6Al4V) used for foil is just too tough. My experience has been that putting Al sheet around it rarely works because the Al is much softer and cannot support the Ti. Putting Ti sheet around it might work with carbide dies, but not with the steel ones which come with the unit. They just get damaged (and I had ro reface them).
Yes, sometimes a paper punch will work, because the clearances on those are much smaller (because paper is much thinner). How long the punch will last - that's another question.
So how does one do this commercially for Ti foil? Laser cutting is often used.