Is it possible to fit data to a continuous distribution when distribution parameters are estimated?

I have read this and this, but some things are still not clear to me, thus posting this question. My apologies if I use wrong terminology.

I am going to conduct Monte Carlo simulations and parameters I need to sample in MC simulations so far don't have any known distributions. Thus I need to get these distributions from the real data.

These data are continuous (measurements), and I would like to describe these data with (preferably) parametric distributions. There is no general opinion how these data might be distributed.

I know it is possible to use estimated non-parametric distributions (using kernel density function) for Monte Carlo, however I do want to first try to fit data to parametric distributions since:

1. It is easier to communicate/share with the community results with parametric distributions.

2. Before using kernel density function to define the pdf I need to check that data didn't fit any of known parametric distributions.

Based on what I have read, I make a conclusion that if data are fit to a distribution and distribution parameters are estimated from data, there is no single test, that would allow to accurately determine which distribution(s) are most likely to describe the data.

If my conclusion is (mostly) correct, what would be your way of describing data for Monte Carlo simulations? If I am wrong, which test in your opinion should serve the goal?

Note: things I have tried to do so far:

1. Plot histogram of data and plot all continuous distributions on top of it. Distributions were obtained by fitting data (i.e. using estimated parameters).

2. Fit data to distributions and run Kolmogorov-Smirnov test to check KS statistic and p-value.

3. Calculate quantiles for a probability plot and plot it (scipy.stats.probplot python function). From documentation it Generates a probability plot of sample data against the quantiles of a specified theoretical distribution

I didn't get any positive identification of distribution(s) that are likely to describe data.

For Item 1 it is hard to just visually decide what fits better.

Item 2 didn't return any p-value that I would consider as enough to accept the null hypothesis (data are sampled from a given distribution).

Item 3 didn't show any sufficient coincidence between the line for theoretical distribution and data (tails are always far away from the theoretical).

On stackoverflow I got advice to try chisquare test with binned data, but have not tried yet.