Einstein wanted to omit the unnatural second term on the right-hand side

I was reading the following topic on Wikipedia but couldn’t understand few points: https://en.wikipedia.org/wiki/Mass–energy_equivalence#Mass–velocity_relationship. I’d appreciate it if you could keep the answer simple.

I have highlighted the confusing parts in this picture.

Please notice the second term on the right "$m_{o}c^{2}$".

$$E_{k}=m_{o}c^{2}left( frac{1}{root{2}of{1-frac{v^{2}}{c^{2}}}}-1right) =frac{m_{o}c^{2}}{root{2}of{1-frac{v^{2}}{c^{2}}}}-m_{o}c^{2}.$$

The Wikipedia says, "Without this second term, there would be an additional contribution in the energy when the particle is not moving."

That additional contribution in energy would come from the rest energy which is huge. In other words Ek would equal rest energy when the particle is not moving. I don’t see the purpose of that Wikipedia statement.

I find the following Wikipedia statement the most confusing, it says, "Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the energy at rest zero."

In my humble opinion, if the second term $m_{o}c^{2}$ is removed, it messes up everything. $E_{k}$ equals rest energy when the mass is not moving which doesn’t make sense as I said earlier. Above all, in absence of the second term on right, when mass is moving its kinetic energy would consist of the intrinsic potential energy as its part as well. You need the second terms to remove the intrinsic potential energy from kinetic energy. Why would Einstein think of removing it?