JR Pass -- Worth it?

I am considering purchasing a 21 day JR pass (I require this duration to travel in Japan for some events).

At a minimum, I will be travelling:

1.Tokyo-Kyoto return
2. Tokyo-Hakodate return
3. Tokyo-Kyoto return a second time.

And then whatever local transportation I can cover with the JR pass.

When I use the JR Rail Pass Calculator on the Japan guide, I am told the pass is not worth it. It calculates my total costs for these three trips as 92,000 Yen. 40,000 of this is the Tokyo-Hakodate return trip. And it states that only 53,000 yen is covered by the pass. When I enter only the Tokyo-Hakodate return trip, it states that only 1,000 yen of this trip is covered by the pass. So the culprit is that trip, it seems.

I don't understand why only 1000 Yen is covered for the Tokyo-Hakodate return trip and why only 53000 Yen of the total 92000 Yen cost of the trips would be covered by the JR Pass. Can anyone explain this to me?

To add to my confusion, when I enter the same itinerary in the JRPass.com calculator, I receive a total cost of 105,040 Yen as total costs with no such restrictions indicated.

Also, I will be in Tokyo for much of the 21 days and traveling extensively throughout the city. Practically speaking, how much would one save on local Tokyo travel with a JR pass.

Finally, I would like to know people's experiences of how any restrictions of the pass affect travel between Tokyo-Kyoto. For example, I read that one requires a supplementary ticket at 4960 yen to take the Nozomi train. It seems like this added cost for the Nozomi train alone would make the pass not worth it in my case.

Thank you for your time.

Your question raises several issues. Plus I didn't see what you saw, but saying for Tokyo-Hakodate only 1000 yen worth is covered by the pass is totally nonsensical. All of it by JR shinkansen is covered, and if you look up the fares online, I find the fare one way totals 23,630 yen from Tokyo to Shin-Hakodate-Hokuto Stn.
By my calculations, you are just barely breaking even with your long distance plans. Though it's still unclear why you're going Tokyo-Kansai round trip twice. Local travel on JR trains around Tokyo adds a little, but in the big picture doesn't add to much. Not to mention that while you can go to many popular places in Tokyo by JR, you certainly can't go to all, and subways provide much better coverage. There are good 48 & 72-hour passes where you can save some money, depending on how much you get around.

But just because you break even does not mean that the rail pass is the best way to go. The train to Hakodate will take you about 4 hours 15 minutes with the best connection. But you can fly up there much faster, and at less than half the cost. JAL has its Explorer Fare with many flights only about 11,000 yen one way for foreign tourists. Plus the airports are both inside the cities you're going to. Even adding in the times to get to/from the airports, it is much faster. You could use the same fare for the Tokyo-Kansai trips as well, and fare for those are around 7700 yen one way - again, way cheaper than the train.

Also I suggest you don't follow any vendor's rail pass calculators - I've seen them pad the numbers to make it look like a pass is more attractive. Some add in the reserved seat charges as part of the fare, but this is not necessary most of the time - I usually travel in the non-reserved cars, and I have almost never had a problem getting a seat in many decades. Of course if you take a Hayabusa train to Hakodate, seat reservations are required. But it's still a bit of trickery and slight of hand.

Don't use a calculator - look up the actual fares on a real website. Navitime is a good one, you can specify a rail pass in the routes.
Off the top of my head, Tokyo-Kyoto return is 25k and to Hokkaido 40k return is a reasonable estimate so 90K total - close to the 100k pass cost.
But, if you are only saving a small amount with the pass and not really doing a lot of other travel, single tickets can be a much better and convenient option. I recently did a trip - 92,000 of trains per person across two weeks and while the pass might have saved some yen, the extra time booking tickets, transfers, and travel time were simply not worth it for me as I was able to guarantee seat bookings well in advance (trip was as far north as Aomori and as far south as Hiroshima - I had budgeted 90k pp).

Yup, the calculator is wrong. It claims a Kyoto to Shin-Osaka round trip would cost 1000 yen on the Shinkansen. This is completely inaccurate, and is closer to 5000 yen each way. Locals tend not to use the Shinkansen for this trip as it is too costly, trust me if it cost 1000 yen for a roundtrip the Shin-Osaka to Kyoto bullet train would be rammed on every trip. I'd suggest 500 yen each way is the price for the snail rail option.

1000 yen is the rough cost of a round trip by regular trains between Kyoto and Osaka. Normal people will not take the shinkansen to do the round trip between Osaka and Kyoto, and reasonable rail pass calculators, like the one linked below, would not compare the cost to a shinkansen round trip:
https://www.japan-guide.com/railpass/