Client doesn't return when you create a client class ROS2

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Hi community, I would like to write a client and server node using classes. I started writing the client in a class "form" using the tutorial in the documentation and I used the same messages and the same identical code for the server node. I wanted to create a client class which is querying each 500ms the same msg to the server. Searching on the ROS Answer I found a similar question. And using a solution like that I got that the client now is stuck in a deadlock or similar because send the first request but later no and the code continue to run.

The client that I had written is this:

#include "rclcpp/rclcpp.hpp"
#include "example_interfaces/srv/add_two_ints.hpp"

#include <chrono>
#include <cstdlib>
#include <memory>
#include <functional>

using namespace std::chrono_literals;


class client_foo : public rclcpp::Node
{
    public:
    client_foo(const std::string &node_name) : rclcpp::Node(node_name)
    {
        client_ = this->create_client<example_interfaces::srv::AddTwoInts>("servizio", rmw_qos_profile_services_default);
        cb_grp_client_ = this->create_callback_group(rclcpp::callback_group::CallbackGroupType::MutuallyExclusive);

        while (!client_->wait_for_service(1s))
        {
            if (!rclcpp::ok())
            {
                RCLCPP_ERROR(get_logger(), "Errore mentre aspetto");
                exit(1);
            }
            RCLCPP_INFO(get_logger(), "Servizio non disponibile aspetta...");
        }

        timer_cb_ = create_wall_timer(500ms, std::bind(&client_foo::start, this));
    }

    private:
        rclcpp::Client<example_interfaces::srv::AddTwoInts>::SharedPtr client_;
        rclcpp::callback_group::CallbackGroup::SharedPtr cb_grp_client_;
        // Timer per il callback
        rclcpp::TimerBase::SharedPtr timer_cb_;

        example_interfaces::srv::AddTwoInts::Response::SharedPtr start()
        {
            auto request = std::make_shared<example_interfaces::srv::AddTwoInts::Request>();
            request->a = 1;
            request->b = 2;

            auto result = client_->async_send_request(request);
            result.wait();

            RCLCPP_INFO(get_logger(), "Results!");
            return result.get();
        }
};


int main(int argc, char *argv[])
{
    rclcpp::init(argc, argv);
    rclcpp::executors::MultiThreadedExecutor exe;

    std::shared_ptr<client_foo> client_node =
        std::make_shared<client_foo>("client_foo");

    exe.add_node(client_node);
    exe.spin();
    return 0;
}

Probably the question is very silly but I don't get the point where the code fails. Someone can help me?

Thank you so much.