if I calculated the CountHat for each variety at each time point and subtracted that from the actual point and squared it that would give me the MSE basically, (y-ŷ)². From there I can calculate the SE by simply taking the square root.
I'm afraid not. What the R non linear routine is doing is taking some initial values for a, b and c, working out the yhat or CountHat for each point (incidentally, how did you get that hat in (y-ŷ)²?), then subtracting and squaring as you have suggested, and adding to get SSE (sum of square errors). Then it mucks about with the values of a, b and c until SSE is the smallest possible value. Then it prints a, b and c and calculates t50.
You want to compare the various t50s so you first need to know how accurate each estimate is, that is you need the SE of each t50. In some special simple situations there are formulas for the SE. Not here, alas.
You do have estimates of the SEs for a, b and c done by the jackknife method, but I have to say I don't trust them much but even so. they will be correlated and can't be used to give you SE t50.
You can do your own jackknife estimation for SE t50.
Take Aurora with say 12 data points. You already have found t50 for the full data set.
Remove point 1. Do the same regression on the reduced 11 number set. Find a, b and c and so t50#1. Record t50#1
Replace point 1 and remove point 2. Do the regression on the reduced 11 number set. Find a, b and c and so t50#2. Record t50#2
Replace point 2 and remove point3. Get t50#3. And so on to t50#12. Miss a different one out each time.
Find the SD of the 12 T50s and multiply by sqrt(11^2/12). This is your SE t50 for Aurora.
Whew.
https://www.statisticshowto.com/jackknife-estimator/ The formula is equivalent to last instruction.
When these are done, you can start to compare the t50s.