Hello Camel.
I have done several calculations in the past to show how long a caravan mover is actually used for when uncoupling and positioning on site, and most times the mover is running for less than 5 minutes. Typically it will use about 1Ah of battery capacity to position a caravan. To put that into a comparison, using a 4A rated televison for 15min would use the same amount of power.
Its all about current times time, and although a motor mover may use quite a lot of current, its only used for very short periods of time. This being the case a good quality 70Ah battery in good condition and with reasonable charge should cope with caravan movers with no problems. You should have no concerns if you use a site with a EHU to keep the battery topped up.
If a problem arises, then its usually the condition of the battery due to age or poor charging.
Motor movers use electric motors. These have a big current demand when you start them up (Stalled current) Your battery needs to be able to supply this, and is indicated by the batteries peak or cold cranking current rating. (Different types of batteries can have big differences in this rating)
This peak demand normally lasts for less than a second after which their current demand drops considerably.
Some mover manufacturers have designed soft start systems to reduce the initial peak current.
When the first caravan mover was was first seen at public exhibitions, a fully weighted trailer was demonstrated for about 30 minuets in every hour the exhibition was open. A 70Ah battery lasted all day and was only changed each morning as a precaution.
I suspect you old battery was somewhat past its best.