The oscillations from Runge's phenomenon are inherent in fitting high-degree polynomials. It's unlikely that any adjustment to your data in terms of spacing or numbers of observations would help much at all.
That's why restricted cubic splines are often used for smoothing. They even work well on Runge's example function (analyzed at even spacing in the x-axis), as illustrated here. Unlike fitting a single infinitely smooth smooth polynomial to cover the entire range of data, restricted cubic splines are a set of individual cubic polynomials joined at a set of knots around which the restriction of complete mathematical smoothness is relaxed. You can specify the locations of the knots, which might be important in your situation given the "hump" in the data near your "time 0" from your previous question.
That said, it's not clear whether smooth curve fitting of polynomials, even with restricted cubic splines, will best address your main research question as described in your prior question:
The hump around the zero point is a policy-related distortion and is my main interest. I use a polynomial function excluding this region to identify what the distribution would look like without this distortion.
As noted in comments and an answer to that previous question, your data would seem to be reasonably fit by an exponential or Weibull function except around the "hump," provided that you choose an appropriate time=0 instead of your shifting the time axis so that the "hump" occurs at time=0. If you are willing to "exclude this region" for polynomial fitting then you can do the same for exponential or Weibull fitting.* That would seem to be both a better and a more defensible approach than high-degree polynomial fitting when you need to present your work to a skeptical audience.
*I'm not sure that this focus on the frequencies of treatment times is the best way to proceed. Looking at your data again, I think you might be better served by a form of survival analysis, focusing on the cumulative incidence of treatment times. That is, starting from a true time=0 (e.g., admission or first encounter with a physician), you examine the fraction of patients who finished treatment before each specified time thereafter. That could smooth out the curve quite well on its own, and such analysis is quite amenable to modeling with Weibull hazard functions, of which the exponential is a special case.