Would be helpful if OP provided the specific details of interest to him. Otherwise, we're left speculating as to whether he is worried about a new $75 NK ticket or $10K for EK Apartment !
I have never seen a policy, but cannot say that I have read every policy in existence, which would fully reimburse, without limit, a new ticket when one no shows for that ticket as the result of late arrival on a prior ticket. This is particularly because what is described here are not "connections" and thus not covered as msconnects.
By way of example, the cheapest ticket JFK-LHR on a legacy carrier tonight is $2,774 and typically, the best I have seen is $500 for travel interruptions of 12+ hours. There may be better deals to be had, but that isn't the purpose of the example.
When researching the issue, here are the factors:
1. How much are you saving by purchasing separate tickets?
2. How much extra time are you spending at your transfer point over what you would book on a connection and what is that time worth to you?
3. What incidental expenses will you incur, e.g. hotel and meals?
4. How does the policy define "connection"?
5. If there is coverage at all, what are its conditions, e.g., dollar limits and delay timing?
Don't obsess about CC's. Just because you may need to purchase a specific one-off policy for a specific trip, does not make it a bad deal. Don't get tied up in a CC you don't need to save a few bucks.
Finally, remember that AA does protect on AA-AA and AA-OW separate tickets and that CX protects on CX-CX. Other US carriers may be helpful (stressing "may").