Okay, so what leads you to believe it's a client issue? I can turn your argument against you. I don't get core dumps on my machine. I know others that don't. If it was a client issue, all users would experience it. Your diagnosis is the same as mine. Speculation.
Look at your event logs, specifically:
0a9e372d3b4ad19135b953a78882e789
and
c0000005
Those are Windows errors. Specifically corrupted dll files or core services. Searches on these codes yields users having problems with apps other than FFXIV (one specifically refers to CoD crashing).
Here is what we know:
1.) Not all users are having this issue.
2.) Crash logs specifically point to the problem being Windows crashing and it being core service related.
3.) Researching the error codes provided point not to just FFXIV, but to other apps as well having this problem
4.) Nothing has changed, been installed, or modified on your computer to cause the issue as far as anyone's knowledge.
With the above information, we can then make an educated guess and start narrowing down the problem to get to the root:
1.) It's not an initial client or server issue. If it was, all users would be affected.
2.) Windows seems to be having a conflict it cannot resolve either with it's core services, or interfacing with FFXIV .dll files and another core service (DirectX maybe?)
3.) As other apps have been reported to have the same issues points to a more fundamental scripting or service.
So, It's not client wide. Not a bad load and not a server issue. Error codes are specific on your log point to bad files or conflict (yours specifically). Other programs have this problem as well (heavy rendering programs that use 3D acceleration and higher). Nothing has gone bad/going bad on your machine (that we know of. Diagnostic tools can't pick up intermittent core dumping).
The logical answer would be that your OS, Hard Drive, or GPU are the cause of the issue. Your OS is maintained by Microsoft. Your HDD and GPU by their respective OEM's. It's safe to assume that the client is not to blame.
Try to reinstall The entire client. If that's a no go, try to reinstall Windows. If you have a spare HDD, try that as well just for testing purposes.
I agree with you about my answers being inadequate, but without testing, we won't find an answer. I do this for a living and I know you're frustrated, but please understand that I'm trying to help you.