Pickled Eggs with slightly gooey yolks?

First, even when making pickled eggs at home, to be safe you need to refrigerate them at all times as described at the National Center for Home Food Preservation:

Storing Eggs

After making the eggs, the eggs require some time to season (i.e., pick up the flavors from the pickling brine). Keep them refrigerated at all times. If small eggs are used, 1 to 2 weeks are usually allowed for seasoning to occur. Medium or large eggs may require 2 to 4 weeks to become well seasoned. Use the eggs within 3 to 4 months for best quality.

The closest thing I found that used soft-boiled eggs is this recipe from Serious Eats for Japanese Marinated Soft Boiled Egg for Ramen which also requires refrigeration and also doesn't store for long.

Even if you deem them safe for your personal consumption, it's best not to risk it for other people.

No: if you're making pickled eggs for long-term storage (i.e. in a fridge), you must cook them thoroughly (i.e. hard boil them).

Also, immersing food such as eggs in alcohol and/or acid (i.e. vinegar) will denature the protein, giving it the texture/appearance of having been cooked, but will not kill bacteria/micro-organisms present. Basically, you'll more-or-less end up with hard-boiled eggs after the eggs sit in the fridge long enough, but you're going to put whomever eats them into the hospital.

@DanaBrunson is correct: you can make soft-boiled ramen eggs, but they don't store more than a couple days. If you must have that consistency, you could try making century eggs, which taste OK enough, but not the same as a "normal" soft-boiled egg (and look abhorrent).

I want to do the same thing.

Here's an article about acidification and its effect on pathogens in commercial egg pickling.

https://pubmed.ncbi.nlm.nih.gov/24215686/

It basically says that eggs must be held in refrigeration while approaching the correct acidity of less than or equal to 4.6 pH, but that the bacteria won't actually die until the acidified eggs are allowed to sit at a higher temperature (room, or ambient temperature), after the target acidity is achieved. After about 14 days, no pathogens were detectable, using the methods they outlined, but they knew their exact acidities.

Below a certain temperature, the metabolism of a bacterium slows down to the point that is virtually in stasis, and it needs to be active in order to absorb something that is toxic to it. Most sanitizing and disinfecting agents are designed to work at ambient temperatures for that reason. You kill germs more effectively using the exact concentration of an agent specified by the manufacturer, too. Higher concentrations may make it harder for bacteria to uptake the agent, due to hypertonicity. So, the antibacterial action of many agents depends on the temperature of the solution, the concentration of the solution, and the pH of the solution. This might explain the rationale for specifying that the eggs need to be held at ambient temperature for a period after brining completely under refrigeration.

Another site, dedicated to egg safety, said that whatever cooking method must heat eggs to a uniform temperature of 160 degrees Fahrenheit, or hotter.

https://eggsafety.org/cooking-eggs-right-temperature/

You would need to test your cooking method by temperature measurement before cold shocking them, and then be very careful to always use the exact same process to ensure reproduceable results.

That means measuring an exact amount of water and an exact amount of eggs, placing all of your eggs in at the same time, and shocking them precisely after they are pasteurized to keep them from getting hard yolks.

The December / January 2020 issue of Cook's Country does an excellent breakdown of achieving perfectly consistent egg yolk doneness on pages 24-25.

I can't legally upload images of their copyrighted material, but I can say that they cover the measuring, timing, and cold shocking, and then give nice pictures of egg cross-sections from soft to hard boiled, in one minute intervals of increasing cooking time.

The yolk is the part where the bacteria normally grow the most, so, it's kind of important to achieve a safe yolk temp and then refrigerate during acidification.

There is more at stake than salmonella, too. If done improperly, it is not widely reported, but possible to end up with botulinum toxic in pickled eggs.

https://www.cdc.gov/mmwr/preview/mmwrhtml/mm4934a2.htm

In that instance, the man who was hospitalized had pricked his eggs to speed up brining, but he ended up contaminating the eggs internally.

Another thing to note about botulinum toxin is that it can be broken down by boiling. There was a tragic case in the early 1900s of a family of 12 people who all died of botulism from improperly canned beans. A photo of seven of their caskets grouped around a church altar appears in some college microbiology textbooks.

The editors of the textbooks say that the great shame of it is that they could have survived if the beans had been boiled for 10 minutes before serving.

Still, it is best not to get sloppy with canning or pickling safety, and not to rely upon a final heating before serving to remove potentially lethal pathogens.