Ultimately, the goal is only to make class forcing work. Pelletier's stratification and Friedman's pretameness are what worked for them. Their solutions are well motivated but it is nonetheless a means-to-an-end scenario. So, what does it take to make class forcing work? There are lots of necessary ingredients but two stand out as key ingredients. A good approach is to cook these two key ingredients first and then season to taste.

The first key ingredient is that names should be sets or, more generally, coded by sets. One of the reasons for this is that to make sense of the forcing relation you need to quantify over names and this is not possible in ZFC with proper classes and even in NBG, where proper classes actually exist, you cannot use class quantifiers in instances of comprehension; MK can get away with a lot more but it still has issues gathering a bunch of classes together in order to manipulate them in various ways. So, the ideal situation is that names are sets but there is a little room for tweaking if you are very careful about it.

The second key ingredient is the maximality principle: if $p \Vdash \exists x\phi(x)$ then there should be a name $\dot{a}$ such that $p \Vdash \phi(\dot{a})$. This is usually accomplished by patching names together. Indeed, if $p \Vdash \exists x\phi(x)$ then there certainly ought to be densely many extensions $q$ of $p$ such that $q \Vdash \phi(\dot{a}_q)$ for some name $\dot{a}_q$. The usual trick is to get a maximal antichain $q_i$, $i \in I$, of extensions of $p$ together with matching names $\dot{a}_i$ with $q_i \Vdash \phi(\dot{a}_i)$ and patch these names together into a name $\dot{a}$ such that $q_i \Vdash \dot{a} = \dot{a}_i$ for each $i \in I$. (Compare with the definition of sheaf in category theory.) In order to make this work, it is best that $I$ is a set and that the axiom of choice holds in the ground model. Again there is a bit of flexibility here. For example, Friedman is able to get by with a slightly weaker version of the maximality principle.

The remaining ingredients often fall into place by themselves, so once you have these two key ingredients most of the work is already done.

Ultimately, the goal is only to make class forcing work. Pelletier's stratification and Friedman's pretameness are what worked for them. Their solutions are well motivated but it is nonetheless a means-to-an-end scenario. So, what does it take to make class forcing work? There are lots of necessary ingredients but two stand out as key ingredients. A good approach is to cook these two key ingredients first and then season to taste.

The first key ingredient is that names should be sets or, more generally, coded by sets. One of the reasons for this is that to make sense of the forcing relation you need to quantify over names and this is not possible in ZFC with proper classes and even in NBG, where proper classes actually exist, you cannot use class quantifiers in instances of comprehension; MK can get away with a lot more but it still has issues gathering a bunch of classes together in order to manipulate them in various ways. So, the ideal situation is that names are sets but there is a little room for tweaking if you are very careful about it.

The second key ingredient is the maximality principle: if $p \Vdash \exists x\phi(x)$ then there should be a name $\dot{a}$ such that $p \Vdash \phi(\dot{a})$. This is usually accomplished by patching names together. Indeed, if $p \Vdash \exists x\phi(x)$ then there certainly ought to be densely many extensions $q$ of $p$ such that $q \Vdash \phi(\dot{a}_q)$ for some name $\dot{a}_q$. The usual trick is to get a maximal antichain $q_i$, $i \in I$, of extensions of $p$ together with matching names $\dot{a}_i$ with $q_i \Vdash \phi(\dot{a}_i)$ and patch these names together into a name $\dot{a}$ such that $q_i \Vdash \dot{a} = \dot{a}_i$ for each $i \in I$. (Compare with the definition of sheaf in category theory.) In order to make this work, it is best that $I$ is a set and that the axiom of choice holds in the ground model. Again there is a bit of flexibility here. For example, Friedman is able to get by with a slightly weaker version of the maximality principle.

The remaining ingredients often fall into place by themselves (provided your target forcing isn't trying to do something unreasonable) so once you have these two key ingredients most of the work is already done.