Glueball,
Your post may be correct for older FDs (& APs) but the majority of newer designs will gradually decrease the command as performance drops to eventually limit you to (say) a minimum of 50 feet per minute in CLIMB mode. Your speed should not decay towards stall unless you're really at the performance limit.
In most modern systems, the FD and AP are all in the same card (or more correctly, pair of cards). The crew set up the Horizontal & Vertical modes on the Guidance Panel and select the NAV source (which could be the FMS) if not in HDG. This is used to produce the FD data. The AP, if engaged, then takes the FD output which is also sent to the displays so that the crew can monitor it (or,of course, use its guidance to fly manually).