There is nothing mathematically problematic with either argument, and they are both rigorous and "correct". Yes, you can show that they are mathematically equivalent. The problem is not there. The problem is that the concept of relativistic mass is in itself a source of confusion and does not need to be there in the first place. It is pedagogically a bad idea and so we are hoping that everybody can stop using bad ideas like that.
I will only have rest mass. Everybody is much less confused when we only talk about the one true mass$^\text{TM}$.
The Einstein energy-momentum relation reads as
$$ \left ( \frac E c \right )^2 - p^2 = (m c)^2 \qquad \implies \qquad E = m c^2 \text{ only if not moving!}$$
It is actually rather bad to be considering the energy and momentum as velocity functions multiplied by rest mass, because photons have energy and momentum without rest mass, and this really shows that it is energy and momentum that are the correct independent variables, subject to the above constraint, and that defines the velocity. But we can at least display their relations
$$E = \gamma m c^2
= \frac{mc^2}{\sqrt{1-\left (\frac v c \right )^2}} = m c^2 \cosh \chi
\qquad \bigwedge \qquad \vec p = \gamma m \vec v
= \frac{m \vec v}{\sqrt{1-\left(\frac v c\right)^2}} = m c \hat{\vec v} \sinh \chi$$
You might have seen the relations in front, but maybe not the hyperbolic functions. They are very nice because they naturally satisfy
$$\cosh^2 \chi - \sinh^2\chi = 1$$
which thus makes it very obvious that Einstein's energy-momentum relation is obeyed. It also makes it clear that Lorentz boosts are somewhat of a rotation in Minkowski spacetime.
We can write the energy-momentum as a 4-vector
$$
\begin{pmatrix} E / c \\ \vec p
\end {pmatrix} = m c
\begin{pmatrix} \cosh \chi \\ \hat{\vec v} \sinh \chi
\end {pmatrix} = m ( \gamma c , \gamma \vec v )^T$$
where, if you just divide out the rest mass, then the 4-vector is the 4-velocity, that has the ``always moves in spacetime at velocity $c$" that you saw from Don Lincoln. At normal speeds, every particle is mostly moving in time at velocity $c$, but if you keep speeding up, the particle seems to lengthen its 4-velocity (if you plot it in Minkowski spacetime) and aims more and more to move at 45 degree angle, and there is just no way to get exactly there. You can speak of it as a hyperboloid; it is also obvious from the hyperbolic angle view, which goes on and on without bound.
When you speak of the relativistic mass, that corresponds to $\gamma m$. You can heuristically argue that this thing will grow without bound as you move closer to speed of light. You can go higher in mathematical sophistication by talking about the 4-acceleration, and show the same behaviour, that more and more of the energy given to a system is sent to increasing $\gamma m$ than to increasing $v / c$, and show that it is mathematically equivalent to the above.
But if you start going into those directions, you will run into issues like Longitudinal and transverse mass and quickly pull your hair out. The fact of the matter is that these are all bad ways to look at things.
It is easier to realise that $\gamma m$ is better seen as $E/c^2$, that it has never been the mass that is increasing, but rather the total energy that is increasing without bound. If you just do one division, you will see that
$$ \frac{\vec v}c = \frac{\vec p c}E $$
and then you can either consider things by momentum or energy
$$
\frac v c = \frac{|\vec p|}{\sqrt{(mc)^2 + p^2}} < 1 \\
\frac v c = \frac{\sqrt{E^2 - (mc^2)^2}}{E} < 1
$$
i.e. no matter how much momentum or energy you give to a massive, originally slowly moving object, there is just no way to get them to arrive at the speed of light. You can give it infinite amounts of kinetic energy, and it would just reach the speed of light.
It is just bad habit that we cling on to ideas about mass. Gravitational pull is due to total energy and momentum too, i.e. a hot cup of tea has a greater gravitational pull than the same cup of tea cooled. Inertia, as shown above, is due to total energy. It just so happens that in old physics, we always moved so slowly that the rest mass energy dominated the total energy. For the physics that only cared about energy differences, purely kinetic energy sufficed. Those that used to care about mass, are actually really all caring about total energy. Rest mass only appears in the invariant Einstein's energy-momentum relation, and in the fundamental equations (e.g. Dirac equation). This view is much cleaner, easier for students to understand, and much less confusing, because we have only one thing we call mass, not 4 different things we call mass that we have to remember when to use what.
