I spoke to Arnie's three contacts.
Sharona and Bat-Chen said that the fostering system is through the Welfare Department and therefore whatever rules we came across when applying to adopt would also be applicable when applying to foster. And therefore that they can't help us, and good luck.
Maybe I didn't explain well enough that what I'm looking for is something OUTSIDE of the fostering/welfare system.
I just got off the phone with Gitit and I told her that we would like to be an option for children who need to be removed from their homes in emergency situations. She reiterated that any kind of fostering has to go through the system. But she works (worked?) with a home for at-risk children from which they send children to משפחות אורחות (host families) every so often and that she would look into finding us a child to host. She said, however, that she has three children and she can't imagine how we with five have the energy or desire to take in another. She seemed skeptical and said it is a complicated process and that it might not be the best for our current children and that sometimes the experiences are downright unpleasant.
I told her we want to try.
It's something. And as Arnie says, there are probably all kinds of other off-the-beaten-track options like this that we can find if we just keep looking.
I am still waiting for Noa to get me the contact info for the woman she knows. And beH one day next week I will walk into Summit and see what they say.
My friend Minette also suggested that I try to think of other ways to put my love out into the world. She brought up the idea that I have visisted several times of volunteering with Chibuk Rishon to hold preemies in the hospital who have no one to hold them. I have reached out to Chibuk Rishon once or twice in the past and been told that a) it requires a very serious time commitment, which I was not able to make at the time, and b) they already have more than enough volunteers and are not recruiting more.
I have more time now that my kids are older (and with their father 2-3 days a week). Even if they're not recruiting volunteers right now, maybe there is a list I can get onto and there would be a slot available one day.
Being in that situation also seems like it would be a good way to "slip in" as a home for a baby that ends up suddenly needing one. So this seems like it would be a good option in terms of donating my time and finding a place to pour out the love that I feel I need to give, and also in terms of my goal of some day beH becoming a foster or adoptive parent.
I emailed Chibuk Rishon (firstname.lastname@example.org) and am waiting to hear what they have to say.