Checked luggage and tickets booked separately

This kind of arrangement was complicated and had risks before Covid. If your bag doesn’t make it on the plane with you to FCO—not likely, that airline isn’t responsible to send it anywhere but FCO. Since you are staying overnight at FCO your misdirected bag might have time to be sent to your hotel for your next day’s flight. One way to make this work is to travel with a carryon bag and maximum sized personal item only.

One way to make this work is to travel with a carryon bag and maximum sized personal item only.

Them's fightin' words around here! (I'm a carryon-only guy)
Seems that's the ONLY way to avoid a likely mishap. OP could perhaps ship a box to home address in the States.
"Pack lightly, go carryon-only." Fighting' words to travel by.

Thank you for your wisdom. We have been traveling with a large piece of luggage, but realized that we need about half the clothes and will try to do a practice packing. I'm a pretty experienced traveler, so I should know better. The one absolute at our age is a medical kit.

...maximum sized personal item only.

If the personal item will fit under the seat. Airlines do not guarantee overhead bin space for your maximum sized carry on and can gate check bags if the overheads are full.

Is the NET saving ( after the hotel cost, meals and whatever ) really worth the aggravation of a 2 step DIY travel plan vs. a single ticket from Tel Aviv to the US?

We have a multi-city trip because of a cruise. The return flight from Israel is about 24 hours, so breaking up this very long travel time makes some sense. It would cost us about $1,000 more to schedule an 18-hour flight. We are retired and have flexibility. I'm just looking at all the possibilities. We don't live near a hub so flights are always a bit more complex and never direct.