Since the Jewish concept of rest, doesn't mean, shop, drive, swim, cook etc, many Israelis want an American "Sunday" to shop, drive, swim, cook etc... Those who aren't Shabbat observant according to Torah Laws are actually doing many of these things on Shabbat. They want a longer weekend to do more of it.
When I was growing up in the 1950's New York, there were very strict "Blue Laws" which restricted what businesses could be open on Sundays. It wasn't a shopping day, because the department stores and supermarkets had to be closed. Bakeries and restaurants were open. So were candy stores where I'd buy the giant inky New York Times. Stores owned by Orthodox Jews and closed on Saturday were allowed to be opened on Sunday, but otherwise Sunday was a quiet family day for many.
Today New York, as is most or all of the United States, is a seven day a week place. So many of the laws meant to protect workers and family life have been cancelled by those to promote commerce and all sorts of activities.
Not that long ago in Israel, you could feel Shabbat even in the most secular of neighborhoods. All stores and restaurants were closed; there was no public transportation and almost nobody had cars. Now there are shopping centers davka open on Shabbat, and many people have cars. They certainly won't rest more if Sunday is added to the weekend.
I can't see Israelis working a half-day on Friday and adding hours to the weekday workday to have Sunday off. For the poor, there will be more poverty, because enjoying Sunday the way the legislators and Pro-Sunday Off people envision it will be expensive. Many of the working poor will be working, and their children won't have quality family time. They will find themselves alone, or the parents will spend huge sums of money keeping them cared for and entertained.
This is not my original solution: There is a Jewish way of adding a day off, a "holiday" of sorts to the Israeli Calendar. Rosh Chodesh, the first day of the Jewish Month. Eleven or twelve (not 12 or 13 because Rosh HaShannah is a Rosh Chodesh) holiday days per year would certainly add to the quality of life here. Every month it falls out on a different day, so that it doesn't hit one day and not others. A monthly day off is more festive than the weekly grind.
According to Jewish Tradition Rosh Chodesh is the "women's holiday." I'm sure lots of creative ways can be found to celebrate it.
Pretty soon I'll be going to Tel Shiloh to pray, as I do every month. May this be the last Av of mourning...