A partial answer with some observations about numerical behavior.
You can find the code to compute the numbers in this answer here.
- First we want to define the ratio $R_N$ as the number of ones in primes until number $N$ and the number $N$. If $R_N=1$ and $N$ is prime than we found our solutions (some or all).
- You can see from this OEIS sequence (thanks to @Freddy Barrera) that $R_{10^n}$ is decreasing for n>1. However the delta from $R_{10^(n+1)}$ and $R_{10^n}$ is also decreasing and at some point it can start increasing (or reach a plateau).

! Plot of $R_N$ when N is prime, x axis is the i-th prime, up to the 10k
!
Minimas seem to occur at values just before power of 10, while maximas are around 2*power of 10
Conjecture 1: Base 10 is peculiar and maybe the point on which $R_N >= 1$ (if exists) is at very high value of N. It's time consuming computing primes greater than 10^14 (the value from OEIS) to see patterns. And maybe the bouncing behavior is less marked in other bases.
Conjecture 2: Lower numerical bases maybe give us a clearer $R_N$ pattern.
Let's work in base $2$, that is full of ones!
First we check if we have any solution:
Yes, 3 5 11 19 23 47 61 are, computed up to 10^7 .
Then we want to see if $R_N$ has some patterns:
Plot of $R_N$ when N is prime, x axis is the i-th prime, up to the 10k
Little bouncing, uh? Seems also to tend to a value...
Let's see if 100k give us more clues:
Still a little bouncing.
So, for the moment our best guess is that ${R_N}$ is approaching some value near 0.8
Let's compute $R_{10^n}$ in base $2$:
N ones R_N
10^1 8 0.8
10^2 87 0.87
10^3 889 0.889
10^4 8437 0.8437
10^5 82074 0.82074
10^6 809377 0.809377
10^7 7902408 0.7902408
10^8 78337542 0.78337542
10^9 778137796 0.778137796
Let's see if base $4$ has some similar behavior
Plot of $R_N$ when N is prime, x axis is the i-th prime, up to the 10k
Yes, it is bouncing and tending to some values...
${R_{10^n}}$ base 4:
10^1 3 0.3
10^2 31 0.31
10^3 252 0.252
10^4 2644 0.2644
10^5 25265 0.25265
10^6 217444 0.217444
10^7 2279625 0.2279625
10^8 22415398 0.22415398
Solutions in other bases:
Not found up to 10^7 and base from 3 to 62
My best guess (aka speculation without mathematical proof!):
Even in base $2$ that should be the base with the most number of ones (at least in any given random number) - both because the length of the numbers and the fact that numbers themselves are composed by only $0$s and $1$s - We observe a mostly decreasing trend in the long run, even if little bouncing, leaving me think that no more solution will be discovered after the first 7. In base $10$ this behavior could be even stronger due to the rarity of ones wrt base-2, so $R_N < 1$ for all N and no solution exists at all.
I leave for reference $R_{10^n}$ computed in bases from 3 to 62 pastebin