I have solutions! At least, this is what I found on my Windows 10 PC.
My methodology. I looked at what happened with three browsers: Firefox, Chrome, and Edge. Before a restart, I made sure I was logged into YouTube and Yahoo Finance. I verified that results were consistent by making three restarts with everything I tried. Nevertheless, I can't guarantee that you will get the same results.
- The problem does not seem to exist with Firefox, which has its own cookie system. But...Firefox does have an option to "Delete cookies and site data when Firefox is closed". Also, the same occurs when using private browsing mode. If you use those options, your login cookies will be deleted.
- None of the cookie handling options in Chrome have any effect on deleting cookies on a reboot. Those options are there only to allow you to specify one-time deletions. Edge has fewer cookie handling options (nothing that will help). Note: The current version of Edge is built on the Chrome engine, as are a number of other browsers. What affects one could affect all.
- While looking for ideas, I poked around in the registry and the group policy editor (gpedit.msc). Interestingly, gpedit showed that there can be scripts for Shutdown, Startup, Logon, and Logoff. In checking the relevant folders (gotten to through gpedit), I found that I had no scripts of this type. I used the tool autoruns (from Microsoft's Sysinternals) to get a complete list of startup entities. My attention was drawn to two services, both named Google Update Service (one is gpupdate, the other gpupdatem). I disabled them in services.msc. This did not solve the problem. However, in services.msc I noticed another Google service called Google Chrome Elevation Service. I disabled this one. This stopped the cookie deletion on reboot problem. I re-enabled gpupdate and gpupdatem, and doing so did not change anything. So, it appears that the problematic entity (for Chrome, anyway) is the Google Chrome Elevation Service. I did not test for its effect on Edge. Maybe it does the same thing, or maybe there is something else that performs the same function.
Assuming that what I found applies to the systems of other folks, the responsible company is not Microsoft, but Google. I looked at some online descriptions of Google Chrome Elevation Service. It is supposed to fix the other Google Update Services (and perhaps Chrome itself) if there is a problem in updating. There is no mention of it deleting Chrome's cookies. I suppose that is a hidden "bonus feature."
So what can you do? One choice is to use Firefox. The other is to disable Google Chrome Elevation Service, but if so, you should probably periodically allow it to run in order to assure that you are getting properly updated Google software.
One last thing. Although the problem with Google Chrome Elevation Service shows itself in a reboot, it could be that the service might run while your computer is operational, perhaps when Chrome is closed or you are logged out.