Recently there have been a few papers in this direction. The first one is [AN ON-AVERAGE MAEDA-TYPE CONJECTURE IN THE LEVEL
ASPECT by
KIMBALL MARTIN][1]. The main conjecture 1.1 of that article translated to this setting implies:

**Conjecture:**
   On average $J_0(p)$ has 2 simple factors.

For $p >>0$ the decomposition of $J_0(p)$ always has at least two simple factors since $J_0(p)^+$ and $J_0(p)^-$ are factors of it. So on average Kimball Martin one expects these to be the only two components. In particular for a fixed integer d the conjecture implies that on average $f(p,d) \to 0$ as $p \to \infty$.

Additionally Alex Cowan [has done extensive computations][2] on the decomposition of $J_0(p)$ into smaller factors for all primes $p < 1000000$. And his data only seem to support this conjecture. Indeed his data show the following:

    Total number of simple dimension d factors for p < X

    X       d=1  2    3   4  5  6  Pi(x)
    100000  1686  737 126 35 22 11  9592
    200000  2751 1060 147 36 22 12 17984
    300000  3717 1290 164 37 22 12 25997
    400000  4592 1493 177 37 22 12 33860
    500000  5427 1679 183 37 22 12 41538
    600000  6226 1856 190 37 22 12 49098
    700000  7020 1991 200 37 22 12 56543
    800000  7728 2132 205 38 22 12 63951
    900000  8456 2271 211 38 22 12 71274
    1000000 9172 2412 218 38 22 12 78498

Where `Pi(x)` is the prime counting function. In particular these data seem to be consistent with $f(p,d) \to 0$ on average.

  [1]: https://arxiv.org/pdf/1912.06911.pdf
  [2]: https://arxiv.org/abs/2010.10745