One argument (maybe not of the kind you want) is to use the fact that the wt. 2 Eisenstein series on $\Gamma_0(p)$ has constant term (p-1)/24.

More precisely: if $\{E_i\}$ are the s.s. curves, then for each $i,j$,
the Hom space $L_{i,j} := Hom(E_i,E_j)$ is a lattice with
a quadratic form (the degree of an isogeny), and we can form the corresponding 
theta series $$\Theta_{i,j} := \sum_{n = 0}^{\infty} r_n(L_{i,j})q^n,$$
where as usual $r_n(L_{i,j})$ denotes the number of elements of degree $n$.
These are wt. 2 forms on $\Gamma_0(p)$.

There is a pairing on the $\mathbb Q$-span $X$ of the $E_i$ given by $\langle E_i,E_j\rangle 
= $ # $Iso(E_i,E_j),$ i.e. $$\langle E_i,E_j\rangle = 0 \text{ if } i \neq j\text{ and equals # }Aut(E_i) \text{ if }i = j,$$
 and another formula for $\Theta_{i,j}$ is
$$\Theta_{i,j} := 1 + \sum_{n = 1}^{\infty} \langle T_n E_i, E_j\rangle q^n,$$
where $T_n$ is the $n$th Hecke correspondence.

Now write $x := \sum_{j} \frac{1}{\text{#}Aut(E_j)} E_j \in X$.  It's easy to see
that for any fixed $i$, the value of the pairing $\langle T_n E_i,x\rangle$
is equal to $\sum_{d |n , (p,d) = 1} d$.  (This is just the number of $n$-isogenies
with source $E_i,$ where the target is counted up to isomorphism.)
Now
$$\sum_{j}
\frac{1}{\text{#}Aut(E_j)} \Theta_{i,j} =
\bigg{(}\sum_{j} \frac{1}{\text{#}Aut(E_j)}\bigg{)} + \sum_{n =1}^{\infty} \langle T_n E_i, x\rangle
q^n
= \bigg{(}\sum_{j}\frac{1}{\text{#}Aut(E_j)}\bigg{)} + \sum_{n = 1}^{\infty} \bigg{(}\sum_{d | n, (p,d) = 1} d\bigg{)}q^n.$$

Now the LHS is modular of wt. 2 on $\Gamma_0(p)$, thus so is the RHS.  Since we know
all its Fourier coefficients besides the constant term, and they coincide with those of the Eisenstein series, it must *be* the Eisenstein series.
Thus we know its constant term as well, and that gives the mass formula.

(One can replace the geometric aspects of this argument, involving s.s. curves and Hecke
correspondences, with pure group theory/automorphic forms: namely the set $\{E_i\}$ is
precisely the idele class set of the multiplicative
group $D^{\times}$, where $D$ is the quat. alg. over $\mathbb Q$ ramified at $p$ and $\infty$.  This formula, writing the Eisenstein series as a sum of theta series, is then
a special case of the Seigel--Weil formula, I believe, which in general, when you pass to constant
terms, gives mass formulas of the type you asked about.)