Hi my family is going on a 11-day trip to Japan and deciding between pay-as-you-go or 7 days JR pass or 14 days JR pass has been quite challenging.

Current Plan:
Day 1: Narita to Tokyo
Day 2: Tokyo to Asakusa
Day 3: Tokyo to Shibuya
Day 4: Tokyo to Enoshima
Day 5: Tokyo to Nagaya
Day 6: Nagoya local
Day 7: Nagoya to Kyoto
Day 8: Kyoto local
Day 9: Kyoto to Osaka
Day 10: Osaka local
Day 11: Osaka to Kansai Airport

I think just the shinkansen for Tokyo - Nagoya - Kyoto - Osaka would cost around a 7 day JR rail pass so it might be worth it to buy the 7 day pass and activate it on Day 5.
The main concern I have is that the JR Rail does not cover all train lines (Asakusa for example).
Or having to limit/choose transport option to just JR Rail may make the trip more chaotic than needed. Especially if we are travelling local within the city a lot.

If the savings is minimal, I would rather pay-as-you-go to have some peace of mind.

Can someone help shed some light on this?
Thanks.

Easy answer. Get the 7 day pass and activate it on day 5. Its not expensive to get from Narita to Tokyo and you may even want to take a bus as it might get you directly to your Tokyo hotel. Asakusa and Shibuya are in Tokyo and easy to get to by train or subway. (assuming you are staying somewhere close to a subway or train station) Enoshima is not that far from Kamakura if I remember correctly but I have not been there in decades. Should not be that expensive to get there.

You don't need a JR Pass for that itinerary. Your only significant fares are on days 5, 7, 11. Took a look at hyperdia and those fares were
¥11,300 shinkansen
¥5,700 shinkansen
¥2,910 JR limited express Haruka
Kyoto to Osaka is inexpensive ¥570 on a rapid service train.

That is about ¥20,000 and the 7-day pass is about ¥29,000. I think there is no way that you would spend ¥8,000 on local travel. A 14-day pass would be a big waste of money.

>> I think just the shinkansen for Tokyo - Nagoya - Kyoto - Osaka would cost around a 7 day JR rail pass
Where did you get that idea? A rule of thumb (actual reality) would be that Tokyo to Kyoto is a bit less than half the price of a 7-day pass, so making that roundtrip with a one-way Narita Express would be enough to make the pass worth it. But you are not returning to Tokyo. Really, when I saw your itinerary I thought the easy answer was no.

Your concerns are perfectly valid. There are subways and non-JR railways that would be better options for several of those local trips.

Why are you going to Nagoya? I can't think of a good reason for a family to go there.
Why are you going to Osaka? There are some good reasons to visit, but what are yours? Theme park maybe?

A quick check on the value of a JR Pass would be to find the fare from Tokyo to Kansai Airport on shinkansen and limited express trains. The price should be higher than what I gave above. Probably arournd 1500 more for three reserved seats plus a bit more. The difference between that and the cost of the pass is what you would have to spend on local travel on JR to make the pass worthwhile.

The 14-day pass is a no-brainer no, just looking at the JR fare from Narita Airport to Kansai Airport.

You can take a look at the Rail Passes link at the link below to see if a regionsl pass wouild help. Try the Rail Pass Calculator below that.
Transportation in Japan (japan-guide.com)