In the data I have, in 25,855 PAs, there we 2,465 PAs that began with 0-2 counts. 1,039 ended in a K, 134 ended in walks, and 298 went to 3-2. 379 ended up as a hit. After an 0-2 count, runners reached safely at a rate of .262.
Just for some context:
This 1st pitcher threw 622 PAs, 82 of which started 0-2. 38 ended in Ks, 4 in walks, 10 went to 3-2, 7 ended as a hit, and after an 0-2 count, runners reached safely at a rate of .183.
Arguably the best pitcher I scored for threw 742 PAs, 99 of which started 0-2. 54 ended in Ks, 4 in walks, 4 went to 3-2, 6 ended as a hit, and after an 0-2 count, runners reached safely at a rate of .131.
The only pitcher I scored for that got drafted out of HS was drafted in the 3rd round at 113. He threw 467 PAs, 50 of which started 0-2. 29 ended in Ks, 4 in walks, 8 went to 3-2, 6 ended as a hit, and after an 0-2 count, runners reached safely at a rate of .240.
All 3 graduated at 17. The 1st was 5’9” 135, the 2nd boy was 5’10” 185, the 3rd was 6’2” 185. The 1st was gunned as high as 91, but cruised at 83-85. The 2nd boy was gunned as high as 88, but cruised at 83-85. The 3rd was gunned as high as 93, but cruised at 85-88.
Other than those 3 there were 2,234 0-2 counts in 24,024 PAs. 918 ended in Ks, 122 in walks, 276 went to 3-2, 360 ended in hits, and after an 0-2 count, runners reached safely at a rate of .271.
Since two of those pitchers had the “best” numbers of all the pitchers in the database, and the 3rd was a monster by all other accounts, it’s pretty easy to see that the average pitcher in the database did far worse in those situations. Of course someone else may well have data showing otherwise.