A simple treatment can also be found in Sects. 2.1 and 2.2 of
https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf
The upshot of the heuristics is that you can assume that Newtonian mechanics is (at least approximately) correct in the momentaneous rest frame of the particle. Then via Lorentz invariance you can extend the dynamics to the general case, which leads you to the conclusion that for massive particles you can just use the proper time of the particle instead of coordinate time, leading to the definition of the four-velocity and the four-momentum
$$p^{\mu}=m \mathrm{d}_{\tau} x^{\mu}.$$
since
$$(\mathrm{d}_{\tau} x^{\mu}) (\mathrm{d}_{\tau} x_{\mu})=c^2=\text{const}$$
the "on-shell condition"
$$p_{\mu} p^{\mu}=m^2 c^2$$
holds and thus the relativistic equation of motion
$$\mathrm{d}_{\tau} p^{\mu}=K^{\mu},$$
where ##K^{\mu}## is in general a function of ##x^{\mu}## and ##p^{\mu}##, implies the constraint
$$p_{\mu} \mathrm{d}_{\tau} p^{\mu}=0=p_{\mu} K^{\mu}.$$
That's why only three equations of motion are indpendent, and the fourth follows, and you can as well write the equation of motion for the spatial part of the above manifestly covariant version in terms of coordinate-time derivative, using ##\mathrm{d}_{\tau}=\gamma \mathrm{d}_t##. From this you get
$$\vec{p}=m \mathrm{d}_{\tau} \vec{x}=m \gamma \mathrm{d}_t \vec{x}.$$
and
$$\mathrm{d}_{\tau} \vec{p}=\gamma \mathrm{d}_t \vec{p}=\vec{K}$$
or
$$m \mathrm{d}_t (\gamma \mathrm{d}_t \vec{x})=\frac{1}{\gamma} \vec{K}=\vec{F}.$$
Now of course all the beauty of the covariant formulation is gone, but it's in a similar form as in Newtonian physics and still fully relativistic.
