May the "reaction coordinate" in a chemical "reaction coordinate diagram" be represented by a time axis?

But I didn't find any further explanation why a parametrization by time is ruled out

You won't find that 'further explanation' simply because these energy diagrams are really not about time and few would see it that way.

The diagrams are clear and stylish representations of the energy levels of the various constituents of a given reaction. It reads easier to do this in $text{2D}$ but that doesn't imply the horizontal axis is a time axis.

Edit

I have found several times a representation using time as axis in high school level representations. For example from this page from the ministry of education in Baden Württemberg (Germany)

Sadly, state education (high school) texts are the worst to use as 'adjudicators' here. High school text books are riddled with errors, distortions and misquotes.

Same with 'kahnacademy': a populist educational site, with only one purpose in life; to make money and lots of it too.

Typically studying the progress(yield/conversion) as a function of time is a field known as chemical kinetics where there are basic differential equations of concentration with rrespect to time being equal to a constant multiplied by the concentrations either multiplied together or raised to a power based upon the collision theory mechanism of the reaction. $$ frac{d[concentration Product]}{dt}=K[Concentration Reactant1]^x[Concentration Reactant 2]^y$$ The constant multiplying these concentrations is known as the reaction rate constant K (not to be confused with the equilibrium constant which is to do with chemical equilibrium & thermodynamics). It is actually often confused with this, at equilibrium there isnt even any observable reaction anyway as both the forward and reverse rates are equal. The reaction rate constant is a function of temperature.

Moreover the Gibbs free energy is related directly to the your figures as well $$ Delta G= mu dN + VdP + SdT$$ where $$ mu $$ is a function of both T and P and can infact even be a function of magnetic fields and electric fields. So chemical reactions can be reversed in multiple ways including pressure, temperature, electrolysis and even maybe even magnetism? what are some other ways to do this?