Given the definitions $\vec a_i = \vec F_i/m$ and $\vec F = \sum_i \vec F_i$, and Newton's 2nd law formulated as $\sum_i \vec F_i = m \vec a$, we can prove that all of the following expressions are equivalent:
$$\vec F = \sum_i F_i = \sum_i m \vec a_i = m \sum_i \vec a_i = m \vec a.$$
These definitions together also imply the additivity of acceleration: $$ \vec a = \sum_i \vec a_i.$$
What we cannot do is uniquely derive the definition of $\vec a_i$ (or, for that matter, $\vec F$) from the statement of the 2nd law alone — without knowing what we mean by $\vec a_i$, there's no way to figure that out just from $\sum_i \vec F_i = m \vec a$, since no $\vec a_i$ appears in it.
However, if we take it as granted that accelerations are additive and proportional to forces, i.e. that $\vec a = \sum_i \vec a_i$ and $\vec a_i \propto \vec F_i$, then we can indeed derive from $\sum_i \vec F_i = m \vec a$ that the constant of proportionality between $\vec a_i$ and $\vec F_i$ must, naturally enough, be $1/m$.
Indeed, a perfectly reasonable way to derive Newton's first and second laws would be to start from the following definitions / postulates:
- Acceleration is the change in velocity over time: $\displaystyle \vec a = \frac{{\rm d} \vec v}{{\rm d}t}$.
- When more than one effect accelerates a body, they combine additively: $\displaystyle \vec a = \sum_i \vec a_i$.
- A force acting on a body accelerates it in proportion to its magnitude and direction: $\displaystyle \vec a_i \propto \vec F_i$.
- The ratio of the force and the resulting acceleration equals the mass of the body: $\displaystyle \frac{\vec F_i}{\vec a_i} = m$.
These postulates are almost sufficient to derive $\sum_i \vec F_i = m \vec a$, and indeed all the other equivalent forms given above, except for the fact that postulate 3 implicitly leaves open the possibility that there might be accelerations that do not result from forces.
If we insist on assigning to each component acceleration $\vec a_i$ in our system a corresponding force $\vec F_i = m \vec a_i$ (which we can always mathematically do, as long as $m > 0$), then we will indeed recover Newton's 2nd law as conventionally stated.
However, I would argue that there are, in fact, sound conceptual and pedagogical reasons not to do this, but to instead leave open the possibility of there being sources of acceleration without a corresponding force.
In particular, such "anomalous accelerations" arise naturally in non-inertial coordinate systems, where they represent changes in the apparent motion of an object that actually result from the movement of the coordinate system we're working in. The standard way of handling such pseudo-accelerations involves artificially multiplying them by the mass of the object, and calling the result a fictitious force, but they aren't really proper forces at all — for one thing, they don't generally obey Newton's third law.
I would argue that, conceptually, it makes a lot more sense to treat these effects as (apparent) accelerations not arising from any force, than to artificially invent imaginary pseudo-forces to attribute them to. Not only is this approach simpler (in terms of not postulating unnecessary entities), but it works better for numerical calculations (where avoiding the needless multiplication and subsequent division by $m$ simplifies the arithmetic and reduces rounding errors) and allows the equations of motion to be naturally applied even to inertially moving objects of unknown or zero mass ("tracer particles"), without having to take awkward limits as $m \to 0$.
A notable special case is gravity, which is a real force in Newton's physics, but a pseudo-force in general relativity. Even when working in Newtonian physics, it can be numerically convenient to treat gravity as a pure forceless acceleration, rather than as a force proportional to the mass of the object it acts on. Indeed, I expect that there's quite a lot of naïvely written physics code out there whose speed and numerical accuracy would be improved (marginally, at least) if programmers were taught to treat gravity as an acceleration and not as a force, and thus avoid unnecessary multiplication and division.
Ps. For a historical perspective, it might be interesting to look more closely at Newton's original formulation of his second law (via Wikipedia):
"Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur."
"Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd." (Motte 1729)
Taken literally, this form of the second law may be written mathematically as $\frac{{\rm d}\vec v}{{\rm d}t} \propto \vec F$, or, to use Newton's own notation for time derivatives, $\dot{\vec v} \propto \vec F$. (An even more literal reading might skip the derivatives altogether, and interpret Newton's statement here merely as $\Delta \vec v \propto \vec F$.)
It's worth noting that Newton here merely talks about a single "motive force"; while the following commentary notes that subsequent changes to the velocity are additive, this statement of the second law makes no explicit mention of how multiple non-parallel forces acting simultaneously should combine.
Nonetheless, it is remarkable how closely this original form of Newton's second law resembles what I stated as my postulate 3 above (given that I hadn't actually looked it up before writing this addendum). Indeed, by giving a name — "acceleration" — to the "alteration of motion" over time, and noting that multiple accelerations combine the same way whether they are applied consecutively or simultaneously, we essentially obtain the postulates 1–3 above; all that remains is the explicit constant of proportionality given as postulate 4 above.
