When studying demographics, you often need to account for the population size of a city or region (GDP per capita, is a good example).
Normalizing by $10^6$ people is a nice round number, but lots of cities and regions don't have that many people.
So if we want to keep numbers in the range [1, 100], 1M is too big a denominator.
That's why a lot of studies use $10^5$ people as a scaling factor.
The same goes for RNA-seq and other sequencing measurements.
Your readout will typically contain $10^7 - 10^8$ reads for a bulk sequencing run.
If your sample has thousands of cells, this means very few cells will have > $10^6$ reads assigned to it.
But the TPM unit was designed for bulk measurements and that's what most people are used to.
So we can keep the same idea of scaling by number of reads, but just use a different convenient scaling factor.
This way we get both scaling by sequencing depth and nice numbers to work with.
As for your $\log(1+x)$ comment, yes, if $x$ is in TP10K, the low read counts will be biased more than if $x$ was in units of TPM.
But consistently lowly-expressed genes often aren't of interest.
So this won't be too much of a problem.
The bigger issue is the zero count over-inflation that is common in scRNA-seq measurements.