AMD being competitive at the time is what mattered, but there's still technological advancement needed for them to be competitive. In this case, it was AMD using copper interconnects that allowed them to not only hit 1 GHz, but quickly clock up from there: https://en.wikipedia.org/wiki/Athlon#Original_release

In terms of marketing, the "GHz" barrier was special, because surpassing it has indeed created a lot of recognition in the general public for the fact that the AMD Athlon CPUs were better than the Intel Pentium III CPUs.

In reality, of course what you say is true and the fact that Athlon could previde a few extra hundreds of MHz in the clock frequency was not decisive.

Athlon had many improvements in microarchitecture in comparison with Pentium III, which ensured a much better performance even at equal clock frequency. For instance, Athlon was the first x86 CPU that was able to do both a floating-point multiplication and a floating-point addition in a single clock cycle. Pentium III, like all previous Intel Pentium CPUs required 2 clock cycles for this pair of operations.

This much better floating-point performance of Athlon vs. Intel contrasted with the previous generation, where AMD K6 had competitive integer performance with Intel, but its floating-point performance was well below that of the various Intel Pentium models (which had hurt its performance in some games).

There was a time where increased clock speeds, or more generally increased processor throughput was important. I can remember when computers were slow, even for things like browsing the web (and not just because internet connection speeds were slow), and paying more for a new faster computer made sense. I think this time period may well have lasted roughly until the "GHz era" or thereabouts, after which even the cheapest, slowest, computers were all that anybody really needed, except for gamers where the the solution was a faster graphics card (which eventually lead to GPU-computing and the current AI revolution!)