I bisected the changes I made to the source code in the broken workspace and I finally pinpointed the culprit.
I lightheartedly added these dependencies in my pom.xml to provide a JAXB implemenation to a third-party package I must use in my project:

    <dependency>
        <groupId>com.sun.xml.bind</groupId>
        <artifactId>jaxb-core</artifactId>
        <version>2.3.0.1</version>
    </dependency>
    <dependency>
        <groupId>javax.xml.bind</groupId>
        <artifactId>jaxb-api</artifactId>
        <version>2.3.1</version>
    </dependency>
    <dependency>
        <groupId>com.sun.xml.bind</groupId>
        <artifactId>jaxb-impl</artifactId>
        <version>2.3.1</version>
    </dependency>

Those dependencies do not play well with any JDK 11+, and they seems to silently break the Java Language Server. If you ever find yourselves in a situation like mine, use these instead:

    <!-- API, java.xml.bind module -->
    <dependency>
      <groupId>jakarta.xml.bind</groupId>
      <artifactId>jakarta.xml.bind-api</artifactId>
      <version>2.3.3</version>
    </dependency>

    <!-- Runtime, com.sun.xml.bind module -->
    <dependency>
      <groupId>org.glassfish.jaxb</groupId>
      <artifactId>jaxb-runtime</artifactId>
      <version>2.3.9</version>
    </dependency>

You can disable the Language Support for Java(TM) by Red Hat extension in VS Code.

It can either be disabled entirely, or just for your current workspace. If you never want to use it, you can uninstall the extension instead.