How to make chewable silicone beads?
I've used a food grade silicone in the past that has the consistency of pudding. It is rather soft and very weak and would not hold up in the example you provide. I have used a more expensive silicone, also food grade that requires substantial mixing to get good results, often requiring vacuum degassing for smooth surfaces. In your example, I expect that small surface bubbles may not be a problem.

In order to ensure my information is correct, I did a quick search for food grade silicone and discovered that Smooth-On is still in business and provides various degrees of Shore Hardness levels.

The second link shows that Smooth-On 960 has a Shore of 60A, which equates to the hardness of a typical automobile tire. That's pretty hard, but that's also an opinion.

I have a silicone mold in hand, which can be gripped and torn, but I don't recall the hardness figures, making it only slightly useful as an example. A spherical part produced from a mold will be more difficult to tear as it is going to have fewer gripping points (none) unless otherwise damaged.

A search for binding agents results in products used to release parts from molds, not useful in this case.

Regarding your mold making question... silicone sticks only to itself. You could make the mold from anything other than silicone and pull the beads after it cures. I have a lead shot split mold that is a series of cavities with tiny funnels into which molten lead is poured. That mold could be split open (empty), immersed in the silicone and then closed and set to cure. It's not a recommendation, as I would not consider the mold to be food safe.

If you have a makerspace in the area, or use Craigslist, you may be able to find someone willing to 3D print a mold from food-safe filament. The mold could be akin to the lead shot split mold but far safer for use in the manner desired.