Bein Hazmanim definitely is important, not because the guys are needed to help around the house, but because yeshiva bochurim need a break every once in a while and by the time pesach comes they are often burnt out from the long winter zman. In the week or two before pesach, there is plenty for the yeshiva boys to do, whether it is shopping for themselves, or helping shop for or clean their house, or even get a job if they need money.
However the idea of in-town yeshivos not starting the new zman until rosh chodesh Iyar is ridiculous. And yes lakewood is in-town. There is no reason that the yeshivos dont start 2 or 3 days after pesach. They had their time to chill out and now its time to go back and continue shteiging.