(Apologies if this was addressed already, but I could not find it.)
In order to use my GPU on Windows 10, I recently uninstalled BOINC and reinstalled it without "Service install", as explained in wiki (https://boinc.berkeley.edu/wiki/Installing_BOINC). From the task manager, it appears that BOINC is not running, and I am unable to start it -- running BOINC manager results (after system prompt about allowing changes to computer) in a message that it is unable to start BOINC client, and I should start the BOINC service. Once I try doing that, I get another message that "Windows could not start the BOINC service on Local Computer" together with "Error 1069: The service did not start due to a logon service." What should I do?

Sounds to me like you didn't fully uninstall BOINC when it was installed as a service.
So let's go that route again.

First go to Windows Settings->Apps and uninstall BOINC from here.

Then start the BOINC installer.
In the third screen of the installer, click Advanced.
Check "Service install" and continue the installation.

When this is done, start BOINC Manager.
In BOINC Manager press CTRL+SHIFT+A to switch to Advanced view.
Then go BOINC Manager->Options->Other options and make sure "Run the client?" is checked.
Check that Enable Manager exit dialog? is checked.
Press Save or OK out of here.

File->Exit BOINC->Make sure to check to stop running tasks and applications->OK.

Back to Windows Settings->Apps and uninstall BOINC.

Start the BOINC Installer.
In the third screen of the installer, click Advanced.
Uncheck "Service install" and continue the installation.

Now after this is done, does BOINC start normally?

Indeed, this solved the problem. Thank you, Jord!

You're welcome. And all those instructions done from memory. ;-)