Yes a load of tosh, otherwise how would have humans evolved if only women who rested lots made 'good' milk? Hunter gather lifestyle isn't really compatibly with sitting around on your bottom given that they are on the move constantly 
It is good advice in one way though, it's showing you just how limited her bfing knowledge is. Now you know not to rely on her advice - you wouldn't take you car to a mechanic who asked you my your automatic didn't have a clutch would you? (I think that's correct - my car knowledge is confined to what DH tells me about cars that I actually retain [i.e. nil!]).
It may be an issue if you have a massive oversupply if you also have very large capacity to store milk (which surprisingly isn't related to cup size) as you may find baby is only able to drink part of the milk stored so the milk is stored for longer than a woman with a smaller capacity. As the longer milk is stored in the breast the lower in fat and the higher in lactose it is, sometimes women with large storage/over supply find their babies have green frothy poos, grumbly/distended tummies, issues with wind (this is because the baby isn't able to make enough lactase - the enzyme that digests lactose - and instead the good bacteria in the gut digest it causing the green poos and the wind/distended tummy. Also it's often associated with massive weight gains unless the baby is refusing to feed due to the 'hit in the face with a hosepipe' effect....
Does any of that sound familiar to you? You haven't mentioned green frothy poos or massive amounts of wind but you have mentioned that baby takes both sides (why would they if they were full to bursting on the 1st), has a sleepy drunk look post feed, seems content between feeds, etc.
Could you ring up on of the bfing helplines to get a RL discussion on your situation or ask for a referral to the Infant Feeding Lead (senior HV in charge of baby feeding - our local one runs all the bfing support groups in the county [in a managerial role] and is training to be an IBCLC)?