Boeing on the other hand has the pilots control the glideslope by adding or reducing power.
On a jet, thrust controls airspeed. Pitch controls path.
The Airbus autothrust is an excellent piece of equipment, and due to the FBW control system there's barely any (noticeable) pitch/power couple as seen in conventional (non-FBW) aircraft with underslung engines. This enables you to happily leave the authothrust engaged all the way down to the deck even when flying with the autopilot switched off. The autothrust will manage thrust to achieve the commanded speed, whether that be pilot selected or FMGC derived. Without going into too much detail, essentially the autothrust will manage the speed in the range from idle thrust to climb thrust; hence the logic of the thrust levers remaining in the Climb detent.
On the flip side, I know that some airlines have a SOP on whether or not you are to flying manually with AT on or off. Do any US airlines allow you to fly the 319/320 without AT during landing?
Neither operator I flew the Airbus for (albeit in Europe, not the US) had any restriction on manual flight, and flight with the autothrust off was encouraged explicitly in the operations manual. Contrary to popular belief, an Airbus can be flown like any other aeroplane!