Is this a correct demonstration for why elements above untriseptium cannot exist?
No, electrons cannot impose any upper limit on the maximum $Z$ of atoms.
The whole research of heavy elements is the research of the nuclei, not the electrons that orbit them. Nuclear physics is about protons, neutrons (or quarks, gluons) and forces in between them and the typical speeds of the constituents are always rather close to the speed of light. Some nuclei classified by $(A,Z)$ are stable, some are short-lived, some are long-lived, some don't exist, and there are islands of stability etc.
For an arbitrarily charged nucleus, however, it's always possible to place an arbitrarily high number of electrons to the orbits.
Special relativity cannot prevent us from doing so and I am confident that people knew that it couldn't since the very discovery of special relativity in 1905. In practice, new quantum mechanics only existed from Heisenberg papers in 1925 but at that time, they already knew that electrons could have been added indefinitely. Since 1928, just 3 years later, they already had the Dirac equation which is enough to study how the motion of electrons in quantum mechanics is affected by speeds approaching the speed of light.
The main reason why the speed of light can't "forbid" some solutions is that relativity simply replaces the electrons that would be "increasingly superluminal" by electrons that are "increasingly close to the speed of light" but subluminal.
We should replace $v$ by $p$, the momentum. The uncertainty principle allows us to estimate the momentum $p$ for a given $Z$ and a given orbit i.e. principal quantum number $n$ etc. Above $Z=137$ or so, the calculated $p$ may indeed exceed $p_0 = m_e c$. However, that doesn't mean that the speed is predicted to be higher than $c$. This claim would be wrong because $p\neq mv$. Instead, in relativity, $$ p = \frac{mv}{\sqrt{1-v^2/c^2}} $$ For an arbitrarily high $p$, we may find a $v\lt c$ for which this equation is satisfied. So if the very heavy nuclei stayed around for a long enough time so that electrons have a chance to fill the orbits to create neutral atoms, they would do so and for a very large $Z$, the inner electrons would simply have speeds that are very close to the speed of light so that the momentum is very high. But the speed would never exceed $c$ and it wouldn't have to.
@LubosMotl Your answer that the electrons are not the limiting factor for the maximum possible atomic number (Z) is correct, but there are a couple of errors in your analysis that I would like to correct.
1) Nuclear physics is about protons, neutrons (or quarks, gluons) and forces in between them and the typical speeds of the constituents are always rather close to the speed of light. True for the low mass quarks, but not for neutrons and protons. These masses are heavy enough and the nuclear binding weak enough that the average velocities are still a relatively small percentage of c.
2) For an arbitrarily charged nucleus, however, it's always possible to place an arbitrarily high number of electrons to the orbits. Not true. You are forgetting that the electrons repel each other and that for a large enough number of electrons the positive repulsion will eventually overpower the attraction of the nucleus. It is true that for a neutral atom the number of bound states is infinite because of the infinite number of possible Rydberg states, but once the number of electrons exceeds Z that is no longer necessarily the case.
Nuclei are like atoms in that there are shell closures that enhance the stability (and binding energy) of certain isotopes. Because of a strong spin-orbit interaction in nuclei the shell closures occur at different numbers than is the case for atoms. These numbers were observed before the importance of spin-orbit effects was known and so they are called magic numbers. Of course there are magic numbers for both neutron and proton orbitals. The number 114 was predicted to be a magic number for protons long before its discovery in 1998. This element is called flerovium. It is missing from the list above because of its earlier discovery but all of the others (113, 115, 117, 118) are members of the island of stability associated with flerovium.
The number 126 is a strong magic number for neutrons (it is the neutron number in the exceptionally stable 208Pb isotope. There are good reasons to believe that it would also be a magic number for protons and so that is probably the next island of stability to be explored experimentally.