The coefficients are stored in writable memory as input data. Only the structure and degree of the polynomial are hard-coded in the program. So if the launch is delayed you just change the data.

Of course, the justification is that compute and memory were at an extreme premium on the spacecraft. Accurately predicting the position of the moon at arbitrary times requires more complicated calculations. You can dramatically optimize the calculation by fitting a polynomial approximation which is valid for the length of the mission. And there's no reason to waste resources fitting that polynomial in space when it could be done on the ground, using much more powerful machines.

The AGC is what tends to get the press. It's easy to forget that the Apollo program relied on a cutting-edge distributed hardware and software stack, most of which was in Mission Control at the Real Time Computing Center.

It wasn't hardcoded. The exact mission parameters (mission clock, moon position, orbital parameters, etc) were uploaded by mission control the morning of the launch. From that point, they had 15 days. The longest Appolo mission was 12.5 days.