A bit complex for such a simple case, but it took me to implement 5 classes, to extract a body from ClientRequest.
It seems to be too much, and I'm still curious if there is a short solution for the problem. You're welcome to propose another answer to this question, so I could accept it.
Unfortunately, it must be said that the design of ClientRequest, BodyInserters and most of the other stuff from org.springframework.web.reactive.*** has a large room for improvement. At the moment it is just a pile of interfaces with tons of methods per each, and it usually takes a lot of efforts to test code, depending on those classes.
The main goal it to get this method working:
static <T> T extractBody(ClientRequest request, Class<T> clazz) {
  InsertionReceiver<T> receiver = InsertionReceiver.forClass(clazz);
  return receiver.receiveValue(request.body());
}
Here is the implementation of InsertionReceiver:
import org.springframework.http.ReactiveHttpOutputMessage;
import org.springframework.web.reactive.function.BodyInserter;
public interface InsertionReceiver<T> {
  T receiveValue(BodyInserter<?, ? extends ReactiveHttpOutputMessage> bodyInserter);
  static <T> InsertionReceiver<T> forClass(Class<T> clazz) {
    return new SimpleValueReceiver<>(clazz);
  }
}
import java.util.concurrent.atomic.AtomicReference;
import org.springframework.http.ReactiveHttpOutputMessage;
import org.springframework.web.reactive.function.BodyInserter;
    
class SimpleValueReceiver<T> implements InsertionReceiver<T> {
  private static final Object DUMMY = new Object();
  private final Class<T> clazz;
  private final AtomicReference<Object> reference;
  SimpleValueReceiver(Class<T> clazz) {
    this.clazz = clazz;
    this.reference = new AtomicReference<>(DUMMY);
  }
  @Override
  public T receiveValue(BodyInserter<?, ? extends ReactiveHttpOutputMessage> bodyInserter) {
    demandValueFrom(bodyInserter);
    return receivedValue();
  }
  private void demandValueFrom(BodyInserter<?, ? extends ReactiveHttpOutputMessage> bodyInserter) {    
    var inserter = (BodyInserter<?, ReactiveHttpOutputMessage>) bodyInserter;
    inserter.insert(
        MinimalHttpOutputMessage.INSTANCE,
        new SingleWriterContext(new WriteToConsumer<>(reference::set))
    );
  }
  private T receivedValue() {
    Object value = reference.get();
    reference.set(DUMMY);
    T validatedValue;
    if (value == DUMMY) {
      throw new RuntimeException("Value was not received, Check your inserter worked properly");
    } else if (!clazz.isAssignableFrom(value.getClass())) {
      throw new RuntimeException(
          "Value has unexpected type ("
              + value.getClass().getTypeName()
              + ") instead of (" + clazz.getTypeName() + ")");
    } else {
      validatedValue = clazz.cast(value);
    }
    return validatedValue;
  }
}
class WriteToConsumer<T> implements HttpMessageWriter<T> {
  private final Consumer<T> consumer;
  private final List<MediaType> mediaTypes;
  WriteToConsumer(Consumer<T> consumer) {
    this.consumer = consumer;
    this.mediaTypes = Collections.singletonList(MediaType.ALL);
  }
  @Override
  public List<MediaType> getWritableMediaTypes() {
    return mediaTypes;
  }
  @Override
  public boolean canWrite(ResolvableType elementType, MediaType mediaType) {
    return true;
  }
  @Override
  public Mono<Void> write(
      Publisher<? extends T> inputStream,
      ResolvableType elementType,
      MediaType mediaType,
      ReactiveHttpOutputMessage message,
      Map<String, Object> hints
  ) {
    inputStream.subscribe(new OneValueConsumption<>(consumer));
    return Mono.empty();
  }
}
class MinimalHttpOutputMessage implements ReactiveHttpOutputMessage {
  public static MinimalHttpOutputMessage INSTANCE = new MinimalHttpOutputMessage();
  private MinimalHttpOutputMessage() {
  }
  @Override
  public HttpHeaders getHeaders() {
    return HttpHeaders.EMPTY;
  }
  // other overridden methods are omitted as they do nothing,
  // i.e. return null, false, or have empty bodies
}
class OneValueConsumption<T> implements Subscriber<T> {
  private final Consumer<T> consumer;
  private int remainedAccepts;
  public OneValueConsumption(Consumer<T> consumer) {
    this.consumer = Objects.requireNonNull(consumer);
    this.remainedAccepts = 1;
  }
  @Override
  public void onSubscribe(Subscription s) {
    s.request(1);
  }
  @Override
  public void onNext(T o) {
    if (remainedAccepts > 0) {
      consumer.accept(o);
      remainedAccepts -= 1;
    } else {
      throw new RuntimeException("No more values can be consumed");
    }
  }
  @Override
  public void onError(Throwable t) {
    throw new RuntimeException("Single value was not consumed", t);
  }
  @Override
  public void onComplete() {
    // nothing
  }
}
class SingleWriterContext implements BodyInserter.Context {
  private final List<HttpMessageWriter<?>> singleWriterList;
  SingleWriterContext(HttpMessageWriter<?> writer) {
    this.singleWriterList = List.of(writer);
  }
  @Override
  public List<HttpMessageWriter<?>> messageWriters() {
    return singleWriterList;
  }
  @Override
  public Optional<ServerHttpRequest> serverRequest() {
    return Optional.empty();
  }
  @Override
  public Map<String, Object> hints() {
    return null;
  }
}