It is important to use careful language to understand what is happening
"What does an oxygen atom favour stability over instability" is a bad way to ask the question and not just because it anthropomorphises the factors causing chemical reactions. The right way to think about the problem is to ask which configurations of atoms and molecules in a dynamic system will have the lowest energy.
Except in a high vacuum atoms and molecules are constantly interacting with other atoms and molecules. When they interact things can happen. Sometimes the atoms and molecules just bounce off each other; sometimes a chemical reaction occurs; sometimes energy is exchanged but no net reaction results. When reactions occur some of them are reversible and some are not.
Since there are many possible reactions occurring in any mixture how do we know what the net result is? The factors that matter are the the energy levels of the possible products in the mixture (I'm simplifying a bit by ignoring entropy) and the degree of reversibility of all the possible reactions occurring.
In the case of a mixture of isolated hydrogen and oxygen atoms a lot of different things can happen (by the way it is very very hard to create such a mixture). One is that isolated oxygens can meet and combine to give oxygen molecules (this releases energy). Or oxygen can meet hydrogen and combine to give an OH radical which can further combine to give a water molecule (releasing a lot of energy). Lots of other reactions can occur. But when the mixture has a lot of water or oxygen in it it requires a large amount of energy to go back to reverse the reaction and generate an oxygen radical. If the mixture doesn't have enough thermal energy to break up a water molecule in some collisions, this reaction is irreversible. It doesn't require any energy input for most of the isolated atoms to react to these products irreversibly.
In a slightly different case, at low enough temperatures a mixture of hydrogen molecules and oxygen molecules will be stable. The molecules banging into each other at room temperature don't usually have enough energy to react or to release the oxygen radical that propagates the reaction leading to water). But it doesn't require a lot of energy input to cause them to react explosively to give water (just enough to break apart a few oxygens or hydrogens to kick start a reaction that will self-sustain because it releases a lot of energy when water is generated.)
In a system as dynamic as a mixture of gases, the possible collisions between the components will explore many possible outcomes. Some of the molecules that result are resistant to further reaction because they sit in an energy well. We call them stable. Isolated atoms don't seek stability, they just happen to react with other things very easily and many of those reaction paths lead to stable molecules. The molecules don't react easily because there isn't enough energy in the system to cause them to break apart (assuming the temperature is not high enough: hydrogen oxygen mixtures are perfectly stable at room temperature unless you are stupid enough to light a cigarette near the mixture, a mistake you will not make twice).
The situation when things are charged is different. The electromagnetic force is strong and causes large forces to exist between oppositely charged ions. These will actively attract each other and so reactions will happen faster than by the normal process of just bumping into each other. Otherwise the same rules apply. But what matter here is that ions might be individually stable as ions but there are strong forces pulling them together with oppositely charged ions into neutral assemblies. What matters is that you can't have, for example, a large clump of chloride ions by themselves. It isn't that the ions are individually unstable, it is just that strong charges produce strong forces attracting opposite charges.
In summary, when we say that something like an oxygen atom isn't stable, we mean that in the normal course of events oxygen atoms can combine irreversibly and very easily into molecules that are stable (which in turn means they have lower energy and sit in a potential well that is hard to escape from). No molecule or atom in a gas mixture seeks anything, but the statistics of molecular collisions will explore every possible potential well in the space of possible products and the deep wells will be full because the isolated atoms fall into them very easily.