Are elementary particles with inertial mass actually composite particles? In the standard model, electrons, muons and taus are elementary particles i.e. they aren't composed of more fundamental particles. But after watching this YouTube video animated by an actual scientist, several statements are made starting at 9'20", which don't entirely mesh with my current understanding of the standard model. From what I understand, the host says:


*

*A particle "has" 2 kinds of mass: the gravitational mass acquired through gravitational interactions with matter and the inertial mass acquired through interactions with the Higgs fields.

*Elementary particles don't have any inertial mass. That's why they don't experience time and can move at the speed of light.

*Particles that can decay or oscillate between states e.g. electron's chirality flip are composite particles.

*A massive electron is actually composed of a left-handed electron and a right-handed anti-positron, which are both massless on their own and so are the real elementary particles.


My conclusion here is that muons and taus, which can decay, also aren't elementary particles.
Although he didn't mention it, I'm pretty sure he's not talking about the standard model. So about which model is he talking about? Or am I missing something and I misunderstood everything?
The YouTube video is from the PBS Space Time channel and is titled "The Origin of Matter and Time", released 27 Jan 2016. Its host is Matt O'Dowd, an astrophysicist.
 A: I think that the video is intending to address the standard model.  However, I got confused at about 9:40 or so, right about the same time you did.  Although the video presents an overall correct view of relativity, I think that the editors of the video need to re-write the segment beginning at about 9:40.  In short, the video seems to have a few facts incorrect towards the end.
It is difficult to pinpoint exactly where the facts get fuzzy, and it is difficult for me to pinpoint what exactly is being said that is wrong - the narrator's explanation is entirely qualitative at that point, and entirely qualitative explanations like this are difficult to critique for accuracy, irrespective of whether or not they are correct.
But at any rate, it might help a few things if I give a point-by-point comment on your four points that you list:
Point 1:

A particle "has" 2 kinds of mass: the gravitational mass acquired
  through gravitational interactions with matter and the inertial mass
  acquired through interactions with the Higgs fields.

Whether or not the inertial mass is the same mass as the gravitational mass (in every way) is ultimately an unresolved issue.  All astrophysical measurements of this question indicates that they are the same insofar as our instruments can tell, but high-energy physicists still like to ponder the topic.  My point in this is that I encourage you to not get carried away thinking that they are measurably different things.
Point 2:

Elementary particles don't have any inertial mass. That's why they
  don't experience time and can move at the speed of light.

This is one part that the video you mention just has their facts flat out wrong.  On the one hand, high-energy physics does distinguish between "bare mass" (before any sort of higgs-coupling has happened) vs. invariant mass (the particle rest-mass that we observe in a laboratory).  However, even with this distinction, no one is really claiming that any fundamental particle is massless (except for the photon, of course).
Point 3:

Particles that can decay or oscillate between states e.g. electron's
  chirality flip are composite particles.

For the sake of simplicity and brevity, I'm going to say that oscillating or decaying into other particles doesn't really connect with being a fundamental vs. composite particle.  One could just as well think that a fundamental particle "decays" by annihilating itself and leaving two new particles in its place.
All of the above rebuttals make point #4 a moot point.
Enjoyed your post!  Thanks.
A: There are a lot of misconceptions here:

A particle "has" 2 kinds of mass: the gravitational mass acquired through gravitational interactions with matter and the inertial mass acquired through interactions with the Higgs fields.

Mass is not acquired , it is inherent in the four vector describing an elementary particle or the four vector of a composite of elementary particles.
The mass acquired at the symmetry breaking time in the generation of matter in the universe is unique and not variable and characterizes the particles.

Elementary particles don't have any inertial mass. That's why they don't experience time and can move at the speed of light.

The concept of inertial mass is out of use in particle physics, though it exists as the "relativistic mass" quantity. For a particle (or a system) at rest it is the same as the rest mass or invariant mass, i.e. the "length"  of the four vector describing the particle (or system of particles).
This is the table of  elementary particles, they are represented by a point, not a volume, and most of them have rest mass and cannot reach the velocity of light:


Particles that can decay or oscillate between states e.g. electron's chirality flip are composite particles.

Wrong, they flip but they still are point particles with mass.

A massive electron is actually composed of a left-handed electron and a right-handed anti-positron, which are both massless on their own and so are the real elementary particles.

This is completely confused and wrong. An electron is an electron and can be measured in the lab.
Well, your summary misunderstands particle physics , I do not know about the video because I am not watching it.
