First off, let's stay humble. To begin with, let us remember that no one really knows the answer to your question. That is the nature of the scientific method:
- We observe macro phenomena.
- We abstract what we see.
- We think, "if that is right, then ... is true".
- We test the extrapolation empirically.
- We refine our abstraction depending on the result.
We never really know. Revolutions in physics occur when some capable and questioning individual thinks there is a hole in the status quo and proposes a solution. He/she is either proven wrong by experiment or goes on to become a giant, usually post humously.
With that in mind, let us remember that the division of the forces into 'strong', 'weak', and 'electrostatic' is an entirely artificial abstraction that has proven to be useful. It is the current 'standard model' and it is just that, a model. There have been other insights which, as yet, have not revolutionized understanding.
In particular, there is Yukawa's idea of a potential that merges these three forces into a more complex potential. The reasoning is that nature abhors mathematical singularities. Therefore, the electrostatic potential must have a different shape at small radii. Yukawa modelled that idea with a numerator that goes to zero on the inverse radius, but rapidly goes to unity for distances on the order of a nuclear radius. The small radius slope is the 'strong' force, a larger radius transitional slope is the 'weak' force, and the asymptotic slope is the electrostatic inverse square law (the force is the spatial derivative, or slope, of the potential).
With that approach, you get a deep but finite well at the center of a nucleon. With this line of thinking, the prime mover of radioactive decay is entirely electrostatic imbalance. Yukawa's work was, of course, not quite right, but it remains influential. There are many other functions that exhibit the desirable properties, so perhaps we have a case like that of the historical development of the Bose-Einstein statistics underlying black body radiation, where early attempts to reconcile the 'ultraviolet catastrophe' were close, but not quite right.
With regard to the apparent absence of negative charge in the nucleus, it is well to remember that a free neutron will decay into a proton, an electron, and a neutrino wth a half life of about 12 minutes.
This suggests that a neutron can be polarized, since in this line of thinking, it is somehow a combination of an electron and a proton. In fact, this has been tested empirically. I remember attending a presentation of the results of such a test about 15 years ago. At that time, the polarizability of a neutron resisted measurement attempts capable of resolving 10 to the -27 meters, indicating the binding force of that particular combination is very 'strong'. Such a test is 'big' science - it is very expensive.
If you assume these lines of thinking, then in the nucleus the barriers are lower and a electron that unbinds one proton rapidly rebinds another in the stable case. In the unstable case, there are too many neutrons, so there is some non-negligable chance of a beta emission. In the large nucleus case with not enough neutrons there is some non-negligable chance of a whole block breaking off, typically an alpha emission owing to the great stability of that configuration, but in the context of some energetic disturbance, perhaps a fission.
The thinking is that an electron in a neutron can balance more than one nuclear proton owing to the short range shape of the potential.
Finally, radioactive decay is not limited to large nuclei. Get yourself a copy of the isotope tabulation of the periodic chart and spend a few hours (or days or years) examining the relationships between stability, the neutron drip line and the proton drip line. All of that complexity is empirical data and so must be accepted as fact.
Some may criticize the speculative non-standard assumptions underlying this line of thinking. I have the giant shoulders of Yukawa to stand on and help squelch some of that. It is necessary to think beyond the status quo. But more necessarily, one must mathematically quantify the qualitative ideas in a way that predicts known empiricism. That is very hard.
Such boldness will necessarily lead to errors, but it can lead to greatness. Perhaps you are strong enough to answer your own question in your lifetime and become a legend. I know I am not.
What others here have said is a reiteration of the standard model. What they have said is not wrong, but neither does it really address the crux of your question which is rooted in the raw curiosity and wonder surrounding the unknown and unknowable. Kudos for having the courage to wonder!
My favorite quotation of Albert Einstein is:
"It would be sufficient to really understand the electron."
Perhaps the answer to your question is to really understand the neutron, which, in this line of thinking, is the fundamental mechanism governing radioactive decay.