This topic, like in some related threads, lacks definitive knowledge. Whilst I don’t know the 737 in detail, but in the auto-landing aircraft that I do know extremely well there are many factors that have to be considered in order to maintain the required safety levels in comparison with a manual landing. This includes control authority, trim range and rate, and airframe systems – hydraulics, etc. Lack of auto-throttle is just one consideration.
I doubt that any aircraft uses trim bias to enable a landing after an autopilot failure, and as for helping a GA, well 737 pilots report that a manual GA is complicated by the trim bias.
JAR-AWO 123 “Automatic throttle control” requires -
An automatic landing system must include automatic control of throttles to touchdown unless it can be shown that:
(1) Aeroplane speed can be controlled manually without an excessive workload in conditions for which the system is to be certificated;
(2)With manual control of throttles the touchdown performance limits of JAR-AWO 131(c) are achieved; and
(3)The touchdown performance is not critically affected by reasonable errors in speed control.
For inf: Para 131 ‘touchdown performance’ requires that it is improbable that the touchdown is beyond the end of the touchdown zone lighting, 900 m (3000 ft) from threshold;
LEM Glib comments implying that an extra 1000 ft on landing distance doesn’t matter because it is certificated distance, are irresponsible. Although this may be within your limited experience; you may wish to consider acquiring more experience, but hopefully not at the expense of an overrun. You would benefit from reading UK AIC 11/98 “Landing performance of large transport aeroplanes”, which identifies long landings due to float as a contributor to overruns.