Yes, I have a bottle at hand and in the car. And in the guest bathroom there is another one. Especially when I have to visit the doctor/hospital it’s important (bad bugs there). But also in shops etc, doorknobs, pay-machines, pincode readers, elevator knobs, you name it. Public toilets not to mention.
And I wash my hands frequently too.
Sure it helps, that’s a fact. Most germs come via hand-on-hand contact and via food (but you can cook that, not your hands).
I also ask the guests in our house on entering to either wash hands or use the alcohol handdispenser. No one ever objected.
By the way, this is not specifically a good idea for CF. It’s good for everyone.