I try to watch for "sweet spots" in award charts to get us where we need to go as a family of 4 and try to maintain 2 Southwest companion passes in order to make it easy to position for wherever we need to catch our international flights. Sweet spots are, of course, pretty fluid, so I find it's best to have flexible currencies (UR and MR especially since they are both so easy to earn in huge quantities).
Regarding your search with Aer Lingus--it's been a while, but we transferred MR to Avios and booked 4 economy tickets on Aer Lingus BOS-DUB in summer 2014 for my immediate family and 3 tickets on a different flight for in-laws. We also booked 4 biz tickets BOS-DUB summer 2015 with Avios back when it was only 25k Avios per person in biz. On the return in 2015 we wanted to fly home from Shannon and had to fly economy in 2 sets of 2 on consecutive days. Back then I found lots of days that had 4 seats in economy out of DUB, but I think the 4 seats in J were very unusual. Maybe availability has become more limited now? I used ExpertFlyer and set up an alert to find those tickets.
We flew to Bali on Cathay Pacific and mixed up F and J tickets in order to get us all on the same flights (booked with AA miles then, but still possible to book at a good rate through Alaska miles now). Cathay Pacific seems to have great availability in J a year out.
We have always had good luck getting coach seats on United for our family of 4 for overseas travel (very convenient since we're based in Denver) but coach tickets to Europe have been so cheap lately that now we just buy them with our Amex Biz Platinum cards (used to be 50% points back on chosen airline, now it's 35%). MR are so easy to earn and 35% points back on chosen airline and on business class flights is great, because you just have to wait for a sale and then you can get tickets for your whole family. For instance, we have upcoming J travel for our family of 4 to the Cook Islands that came out to 100k RT per person (with no additional taxes or fees.) We'll touch down in the Cook Islands and continue in economy for the 4-hr hop to New Zealand on the same Air New Zealand flight using cheap Singapore Airlines award tickets for the last segment. Our economy trip last year to France only cost 18k miles RT per person with the 50% points back benefit (without any add'l taxes or fees, which I love, since these really add up when buying multiple tickets!) I subscribe to the twitter feed on SecretFlying.com to watch for good deals, but you also look on the premium fare deals on flyertalk.
We also used MR to get our family of 4 to Japan on ANA F this summer while there was a 30% transfer bonus from MR to Virgin Atlantic. It only cost 85k miles w the bonus for each RT 1st class ticket from LAX-NRT using Virgin Atlantic miles. We did have to split into groups of 2, although the flights are only 1 hour apart on the same routing. After a week in Japan, we are using United miles to fly on partner airlines to Australia for only 40k pp for business class (miles required has since increased, sigh, but the ANA deal is still available and I'm sure there will be another transfer bonus. ANA has flights with both F and J, so that is a way to get everyone together or at least get everyone there in 2 groups.)
One final thought is that if you subscribe to blogs, lots of times they'll post when there is unusually good reward availability to a destination. If you are flexible with dates and destinations, have a lot of flexible currency miles, and you're willing to jump on something quickly, you can absolutely book international award travel in J for a family of 5.