As a complete layman, I do wonder why n grows with each subsequent curve, rather than with number of bits. Does this not suggest they were 'searching harder'?

Almost all of the smaller values of n are not valid curve values, so they were searching "harder" but just for valid seeds. If you look at the generator output in [1], you'll see that:

1. Only a very small number of valid seeds were discarded when compared to the number of n values. It seems we don't yet know why about half of the valid seeds were discarded but the explanation of a buggy curve validity checker doesn't seem implausible to me.

2. Directly to your point, most of the prime192 and prime239 curves required larger n values to find a valid seed and they are both smaller and used an earlier timestamp.

[1]: https://words.filippo.io/seeds-solved/