Time-to-event models are probably fine. I think that you have too many age group categories. In my opinion, you can use the results of this analysis to identify better age groupings. That is, which age groups can you combine together? You could also leave age as a continuous variable in the CPH regression model. I would NOT delete the observations for age groups with no events. I would reduce the number of age groups by combining similar groups.
Also, I personally would not define 10-year age groups. I would let the data analysis determine the categories. For example, identify the quartiles (25th, 50th, and 75th percentiles), then create 4 age groups based on those quartiles. Use KM analysis to see if the curves are different between the 4 age groups. Then combine similar age groups.
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