Thanks for pointing this out, Michal!
I was not aware of Spring implementation. Sounds like a language/framework favouring an antipattern. Been there :)
The DI is ok as long as it reduces coupling.
Global state is bad, Global mutable state is a nightmare. It is yet global.
Stateless Singletons are not as bad, buy they are coupled and globals. Therefore, some issues remain.
Concerning Logs, you can always use local context, as you pointed out.
Your notes suggest I should write a second part of this article talking about the severity and partial solutions.
Not all singletons are the same, as you cleverly point out.