Kotlin serialization plugin does not generate serializers in Gradle project

Question

I'm trying to use the Kotlinx Serialization libraries and the associated Gradle plugin. It's supposed to generate serializers for when when I annotate a class with @Serializable, like this:

@Serializable
data class Message(val version: String)

// (...)

Json.encodeToString(Message("1.0"))

It does not work, and the annotation is highlighted with:

kotlinx.serialization compiler plugin is not applied to the module, so this annotation would not be processed. Make sure that you've setup your buildscript correctly and re-import project.

As expected, the code does not compile as I'm expecting to use the Serializers further down the line, with the error:

Type mismatch: inferred type is Message but SerializationStrategy<TypeVariable(T)> was expected

Here is my app-level build.gradle:

plugins {
    id "java"
    id 'org.jetbrains.kotlin.jvm' version '1.6.10'
    id 'org.jetbrains.kotlin.plugin.serialization' version '1.6.10'
}

kotlin {
    jvmToolchain {
        languageVersion.set(JavaLanguageVersion.of(11))
    }
}

dependencies {
    implementation "com.amazonaws:aws-lambda-java-core:$lambda_runtime_version"
    implementation "software.amazon.awssdk:dynamodb-enhanced:$aws_sdk_version"
    implementation "org.jetbrains.kotlinx:kotlinx-serialization-json:$kotlin_serialization_version"
    implementation "org.jetbrains.kotlin:kotlin-stdlib:$kotlin_version"
}

task fatJar(type: Jar) {
    archiveClassifier = 'lambdaCode'
    duplicatesStrategy = DuplicatesStrategy.WARN
    from sourceSets.main.output
    dependsOn configurations.runtimeClasspath
    from {
        configurations.runtimeClasspath.findAll { it.name.endsWith('jar') }.collect { zipTree(it) }
    }
}

I'm using Gradle 7.3.3, and see all this in IntelliJ. I tried launching gradle build from the command line, with similar results, so the IDE does not seem to blame.

I read a lot of similar questions but I could not find an answer that worked for me. Can anyone shed some light on my situation?