I have only skimmed the replies so may be repeating what others have said. This is a structural element, totally different to normal ground bearing foundations. Ready mix is best, then you get a guaranteed mix. The problem is the speed of delivery and getting it in place, if access is difficult then pumping is best.
I can see why the builder wants to use a mixer on site, given the quantities. The questions to know/ask are
1. What strength of concrete is specified
2. What mix is proposed, ie course Aggregate proportion and size, fine aggregate proportion, cement proportion, water content.
3. how do they propose to measure the materials. The proportions are generally by weight, so if doing it by volume then the density of the materials is required and suitable measuring containers required.
4. Using a shovel as a measure is not acceptable.
The other issue is compaction of the concrete and removal of air bubbles. A dense concrete is required which protects the reinforcement, adheres to the reinforcement and provides the required strength. This is usually achieved using a vibrating poker. Hand compaction is unlikely to achieve this. Note, the concrete must not be over vibrated as this can cause the materials to separate.
One check you can do is measure the volume of the beams being cast. Then work out the weight of cement needed and hence the number of bags. Then count how many bags are used. It is best to be there when the work is undertaken.