Why didn't Newton just propose the 2nd Law ($F=\dot{p}$) and leave it at that? The 2nd Law implicitly contains the first, doesn't it? If so, it seems he wasn't following his own Rule #1 of Book 3 of his Principia: "We are to admit no more causes natural things than such as are both true and sufficient to explain their appearances." There are intro textbooks (e.g., this one) that study momentum first and then force $F=\dot{p}$ based on momentum. Cf. also this SE post and comments.
Answer
Why didn't Newton just propose the 2nd Law (F=p˙) and leave it at that? The 2nd Law implicitly contains the first, doesn't it? If so, it seems he wasn't following his own Rule #1 of Book 3 of his Principia: "We are to admit no more causes natural things than such as are both true and sufficient to explain their appearances."
Simply because Newton didn't write the formula $F = m * a$. This was written some time in the middle of the XIX century.
If you want to know what Newton really wrote and the historical back ground and development you can find detailed information here
Newton thought that energy is linearly proportional to velocity. The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "any change of motion (velocity) is proportional to the motive force impressed".
This law, which nowadays is wrongly interpreted as: $F = ma$ (there is no reference to mass here) simply states states: $$[\delta] v > \propto [\delta] Vis_ {motrix}$$ and in modern terms is sometimes (illegitimately) also interpreted as impulse, sort of : $$\delta v \propto J [/m]$$. But mass is not at all mentioned in the second law (as the original text shows) but only in the second definition, where we can see a definition of momentum as 'the measure of motion'
Quantitas motus est mensura ejusdem (motus) orta ex velocitate et quantite materiæ conjunctim = 'quantity of motion' (modern 'momentum') is the measure of the same (motion), originated conjunctly by velocity and 'quantity of matter' (total mass)
and, moreover 'motive force' (vis motrix) is used, like all other scholars of the time, referring to the yet unknown kinetic 'force' that made bodies move, which Galileo had called 'impeto' and Leibniz 'motive power' . The interpretation of this formula as the definition of force in modern usage is an ex post facto historical manipulation.
It was Gottfried Leibniz, as early as 1686, one year before the publication of the Principia, who first affirmed that kinetic energy is proportional to squared velocity (or that velocity is proportional to the square root of energy): $$ v \propto > \sqrt{V_{viva}} [/m]$$. He called it, a few years later, vis viva = 'a-live' force in contrast with vis mortua = 'dead' force: (Cartesian) momentum (mass/weight * speed: $m *|v|$). This was accompanied by a first formulation of the principle of conservation of kinetic energy, as he noticed that in many mechanical systems of several masses $m_i$ each with velocity $v_i$,
$\sum_{i} m_i v_i^2$
was conserved so long as the masses did not interact. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction or in elastic collisions. Many physicists at that time held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum: $\,\!\sum_{i} m_i v_i$ was the conserved kinetic energy........ Full post at the quoted link.
As to the rule you quote it is not his rule but the centuries old Law of parsimony better known as Occam's razor
The words attributed to Ockham, entia non sunt multiplicanda praeter necessitatem (entities must not be multiplied beyond necessity), are absent in his extant works;[21] this particular phrasing owes more to John Punch.[22] Indeed, Ockham's contribution seems to be to restrict the operation of this principle in matters pertaining to miracles and God's power: so, in the Eucharist, a plurality of miracles is possible, simply because it pleases God.[17] This principle is sometimes phrased as pluralitas non est ponenda sine necessitate ("plurality should not be posited without necessity").[23] In his Summa Totius Logicae, i. 12, Ockham cites the principle of economy, Frustra fit per plura quod potest fieri per pauciora [It is futile to do with more things that which can be done with fewer].
His versiom became very popular but the princple dates back to Aristotle (ibidem)
Aristotle writes in his Posterior Analytics, "we may assume the superiority ceteris paribus [all things being equal] of the demonstration which derives from fewer postulates or hypotheses
In the same wiki article you'll see that it is confirmed that Newton's rule was not original
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