The only contradiction from the birthing classes and true aikido would be the actual breathing technique.
For various reasons, the woman giving birth needs to breath shallow. She's delivering a baby, so she needs to keep the stomach muscles tensed up while pushing and even during relaxing stages. If you take deep breaths... it could actually suck the baby back IN... which is the exact wrong direction you want baby to go! Baby needs to get OUT! Shallow breaths keep the oxygen flow up for Mother and Baby... but keeps tension on the stomach.
In Aikido, you need to breath deep breaths to help stay relaxed. This also enables you to remain calm and concentrate on center or one point. Shallow breaths in aikido would wear you out faster and exhaust you.
As for whether or not the art of aikido actually helps a woman give birth? Interesting question! I'd be curious to hear about that myself, seeing as I haven't been a woman for several lifetimes.