What information do you have that makes you think Zibby is going to walk in the summer for nothing? or is this just your hope?
You do realize that the only player that the Rangers really need to sign with this years $13.6M cap number is Shesterkin....right? If you are going to talk about contracts that will be starting in future years at least reference the right cap number for that year.
Kakko and Laf will get bridge deals which will not blow any cap out of the water. They cant demand $7/8M a year as RFA's. It doesnt work like that and they wont have the numbers to justify that for a couple of years anyway.
I think Eichel is a bad idea unless Buffalo takes on salary and doesnt want more than Kakko. Otherwise good luck with his neck Buffalo. There are other UFA C coming due very soon.
Let's say that they give Shesterkin 6m ... they will have roughly 18-20m in space at the end of this season. Strome is for sure gone in this scenario, but Fox's 925,000 is going up to 9m, and you will have to give Blais, Kravtsov and Kakko their RFA deals. Let's assume they all get modest bridges and come in somewhere around 6m total (I suspect it will be higher but I'll lend to your argument) that means that they are going to spend about 15m. Even if they had 22m in cap space, 7m is probably not enough to get Mika, and even if it were, you have to consider Chytil, Laf and Miller are all going to be due for their next contracts the following summer and they have less than 5 million in cap expiring and an extra 3.5ish from their retained contracts/buyouts expiring.
Reasonably speaking, the Rangers if they are competing at the TDL will add rentals with their cap space. If they are not, I suspect Mika and Strome will both get moved due to the upcoming cap crunch. Unfortunately, having Panarin at 11.6, Trouba at 8, and and a 10 million dollar third line is going to cause cap issues for the Rags.
Matthews, Marner, Eichel, Petey, Laine, Nylander, Hischier and many other high performing, top picks, are getting big contracts right out of their entry deal. Yes, it is feasible that Laf and Kakko bridge, especially Kakko who has been relatively underwhelming thus far, but if Laf has a good season and then pots 60 or 70 the following year, there's no chance he is seeking a bridge deal. It is nice to imagine a world where all of your star prospects take bridge deals, but the NHL at large suggests that is not going to happen (Makar contract, wait and see Hughes/Petey with big deals too) ... budgeting 20m for Fox/Laf/Kakko's next deals is probably pretty generous (see above scenario)
If I am the Rangers, I am trying to keep Laf and Kakko and offering something like Kreider/Miller/Schneider/Chytil/2022 1st for Eichel 10-25% retained and going all in this year. You have the space and the assets to go all-in, and you have an impending cap crunch coming (this is the issue with running a rebuild and a retool in the same three years). I think the Rags are stuck with Trouba and Panarin (the latter is not really 'stuck') because using that allure of New York to draw in free agents to then flip them before the halfway point of their contract is a bad look and might prevent them from getting other FAs in the future. Flip Strome somewhere for another defenseman, and trade for a 3C.