The acidic hydrolysis of nitriles yield carboxylic acids as their ammonium salts as you say. The ammonia produced also neutralises the acid, causing the concentration to drop.
Using base provides the salt of the acid, thus making it easier to purify. Adding acid then generates the free carboxylic acid, hopefully pure.
That is just wrong.
RCN + H2O + HCl
RCO2H + NH4Cl (water soluble)
No carboxylate.
Because an amide (intermediate) is more basic than a nitrile, it is difficult to stop an acid catalyzed hydrolysis at the amide stage. It will go all the way to the acid. In this instance, the acid is a solid and likely will precipitate from the reaction mixture. The nitrile is likely also a solid, so it may be more difficult to visually determine reaction completeness or mutual solubility may hamper reaction completeness in a completely aqueous system.
If the hydrolysis is performed under basic conditions, you should end up with a water soluble carboxylate salt. The ammonia is also water soluble and depending upon heat and concentration will be largely in the water layer. It poses not problem. The reason I am more hesitant of using basic conditions is a dianion intermediate is required to eject ammonia. This requires a greater excess of base and more vigorous conditions. Filtration or extraction can remove neutral impurities without adjusting the pH. Acidification will convert ammonia into a water soluble ammonium salt and precipitate the carboxylic acid.
If neutral impurities must be removed by forming a water soluble carboxylate salt, the basic hydrolysis will have one fewer steps. Solubility may determine the route. Under basic conditions, the reaction may be completed without an organic cosolvent, just water. Under acidic conditions, an organic cosolvent may be needed to drive the reaction to completion. This will add to raw material and environmental costs.