But is there an issue with using the GS for guidance on the way down to that level, facilitating a more or less continuous descent?
There is, in many cases. The GS may not be certified beyond the published intercept point, increasing the chances of false captures outside that point.
That said, it is often possible to use GS as a backup descent aid outside the normal guidance limits. For example, it is possible to capture the GS when landing to the south at SCL at over 30 miles out. If it remains stable, it will keep you above all the minimum altitudes on the published arrival. However, it is not technically legal to use the GS for that purpose, and is not safe to use it as sole guidance. Use it as a crosscheck, but use other means for primary navigation.