"My question is, do first order phase transitions ever really exist?"
Yes they certainly do. Most of phase transitions are first order. I would not estimate in percents with confidence, but my feeling is that more than 90% of all phase transitions are the transitions of the first order. That is the answer from the experimental point of view.
Discussing it from the theoreticl point of view, I cannot see, why the fact that the body has a final size brings you to a conclusion that its capacitance should be continuous. The body in thermodynamics is indeed treated as a finite, as soon as one of the thermodynamic variables is, say, volume, or a number of particles. Let me just remind you that the two phases, 1 and 2, may be characterized by their free energies, F_1, and F_2. As soon as the free energy has the variables: V, T and N, it describes a body of a finite volume. The free energies under discussion are two different functions. This is important to understand. In some cases they may be slightly different, but there are certainly transitions where they essentially differ from one another. It depends upon the transition under study. In the transition point, however, the free energies of the phases are equal: F_1=F_2, but not their derivatives. Nothing strange that the second derivatives of these two different functions are not equal. It is, in contrast, not natural to expect them to be equal.
There is a different source of a perplexity in the case of the first order transitions. It is that each experiment is always performed during a certain time, while the transition itself has its characteristic time, or few characteristic times. If the dynamic of the transition is slow, the time of the experiment duration may be not enough for relaxation. It is often the case, if the transition is diffusive, but may be met also in other cases. This kinetic nature may "wash out" the curve C_v=C_v(t) and it may seem to be continuous. This is however, the kinetic effect only.