I have lived in EY for many years without a teudat zehut. It has limited me only in a few things. I use my passport number for most things, including getting a bank account - which you can get at Bank HaDoar if no one else will accept you, but I would also try Bank Discount and the other banks to see what they can set up for you. I have arranged for phone service, electricity, etc, and bought an apartment all without a teudat zehut. It definitely can be done.
Now, as far as Bituach Leumi, there is a law that for a person who has been in EY for one year consecutively (no going back to US for YT), they are considered a "toshav" and they are entitled to get Bituach Leumi and, through that, Kupat Cholim. Definitely look into that. With that, you will get assigned a Bituach Leumi # that will also be your Kupat Cholim # and that number can be used for a few other situations.
My only major limitation that I've found is that Bank HaDoar cannot set up internet access for me to my account because I do not have a teudat Zehut. I don't know what their problem is because other banks and organizations can do it so I don't know why they can't.