This question cuts right to the heart of one of the major difficulties of the shidduch system – the lack of time given to a young man and woman to know each other, have seen each other in a variety of life circumstances from which each can glean how the other would be as a husband/father or wife/mother.
For my part, I took the following approach – when my wife and I were dating (after only ~ 1 month), the first family members we met were my young cousins – ages 10 and 5 – we took them out for a WHOLE DAY – pizza, bowling, park, etc. – and they were as wonderful and rambunctious as any other kids those ages…
By the end of the day, I had a pretty clear idea of how my future wife would interact with children as we dealt with spilled drinks, scraped knees and disappointing results at the bowling alley – she handled all with aplomb!!
However, if the expectation is 3-7 dates and then meet parents and get engaged with limited time together during said engagement, how on earth could one have such experiences?!?!