The "flaps" you mention are called the skirts and the best thing for leather to soften it is any product that contains a good amount of lanolin and beeswax, neats foot oil is not the best thing, it leaves oily residues on the leather and alot of people put it on the grain side (top shiny surface) which is wrong it needs to go into the flesh side (underneath) the leather fibres are more porous this side and all oils should be applied on the flesh side.
Cod liver oil is far superior to neats foot and it leaves the leather with the nice leathery smell too. But any oil alone will not soften leather very much.
When the leather is being tanned, it is stuffed full of fats towards the end of the processing, cod liver oil is used in this process as is tallow.
I find that when hand dressing new hides in my workshop, a small amount of cod liver oil on the flesh side keeps it in tip top condition and a liberal dressing on the grain side of tallow will keep it looking good. Leather loses moisture every day including new leather, whether it is used or not, so keep that in mind when buying new saddlery in a tack shop, it may have been sitting there for ages!
If leather makes a creaking sound when moved it's TOO dry!
If you ever buy new leather with a white greasy looking film or brush strokes on it, it is GOOD as this has been hand dressed with tallow and not a defect as some have thought!
You find good quality English leather with this white bloom on.
My best, by far, dressing I use on old saddlery that needs some tlc is Aussie leather conditioner, it's got the lanolin and beewax and doesn't take alot of work to get the leather nice and supple. £5.50 per tin and it goes along way!
If you can't find anyone who sells it pm me and I will send you some, hope this info helps.