C++ Reallocating pointer to array in dynamic memory in a different function

I have just done a module on pointers and dynamic memory in C++ and am attempting to complete a personal assignment so that I can practice the concepts. The program manages an array of strings that are names. The goal that I set for myself is that the list is stored in the heap (to practice using "new"), and the list is dynamically sized as new names are entered.

Disclaimer: I realize that this is easily accomplished using vectors, and after struggling with this for hours I re-wrote my original code to use a vector for the list with no problems. However I want to learn where my understanding of how pointers work is broken.

The problem that I have with the program is this: I initialize the name array to have zero elements and have a function to add names that handles the dynamic sizing. When first called it seems to re-size the array correctly and add a new name to the the new array. Within the function to add a name, I can print the contents of the new array. I can also re-assign the old array pointer to the address of the new array on the heap. However when I call the print function from main after adding a name to the list, the list does not contain a name. By my understanding, since I'm using pointers I should be updating values directly, so after the add name function terminates, the values should persist. Also, if I attempt to add a second name the program crashes. What am I doing wrong with the handling of memory?

I've searched quite a bit and the closest that I can find for a resolution was this post:

How to make an array with a dynamic size? General usage of dynamic arrays (maybe pointers too)?

I modified my code based on what I understand from that but it still doesn't work properly.

    #include <stdio.h>
    #include <vector>
    #include <iostream>

    using namespace std;

    void add_name_to_list(string * my_list, size_t * list_size);
    string get_name();
    void print_names(const string *const my_list, const size_t *const list_size);

    int main()
    {
        string *name_list_ptr {nullptr};
        name_list_ptr = new string [0];
        size_t name_list_size{0};
        size_t *name_list_size_ptr {&name_list_size};

        print_names(name_list_ptr, name_list_size_ptr);
        add_name_to_list(name_list_ptr, name_list_size_ptr);
        print_names(name_list_ptr, name_list_size_ptr);
        return 0;
    }

    void add_name_to_list (string * my_list, size_t *list_size)
    {
        string new_name{get_name()};
        string *new_string_ptr{nullptr};
            new_string_ptr = new string [*list_size+1];
            // copy existing list into new list
            cout << "List size is " << *list_size << " so *list size == 0 is " << (*list_size == 0) << endl;
            if(*list_size == 0)
            {
                new_string_ptr[0] = new_name;
                *list_size = *list_size +1;
                cout << new_string_ptr[0] << " has been added to position " << *list_size << endl;

            }
            else
            {
                print_names(my_list, list_size);
                for(size_t i{0}; i < *list_size; i++)
                {
                    cout << "At position " << i << " original list is " << my_list[i] << endl;
                    new_string_ptr[i] = my_list[i];
                    cout << "The name " << new_string_ptr[i] << " has been added to position " << i << " of the new list" << endl;
                }
                new_string_ptr[*list_size - 1] = new_name;
                *list_size = *list_size + 1;
            }
            print_names(new_string_ptr, list_size);

            string *temp_ptr{nullptr};
            temp_ptr = new string [*list_size-1];
            cout << "temp ptr is " << temp_ptr << " and my list is " << my_list << endl;
            temp_ptr = my_list;
            cout << "temp ptr is " << temp_ptr << " and my list is " << my_list << endl;
            my_list = new_string_ptr;
            delete [] temp_ptr;
            new_string_ptr = nullptr;
            print_names(my_list, list_size);
    }

    string get_name()
    {
        cin.sync();
        cin.clear();
        string new_name{};
        cout << "\nEnter the full name: ";
        getline(cin, new_name);
        cin.sync();
        cin.clear();
        if(new_name.size() <= 1)
            return "0";
        else
            return new_name;    
    }

    void print_names(const string *const my_list, const size_t *const list_size)
    {
        if(*list_size == 0)
            cout << "The list is empty" << endl;
        else
            for(size_t j{0}; j < *list_size; j++)
                    cout << j << ". " << my_list[j] << endl;
    }

One variation that I've tried based on what I learned from searching is:

    cout << "temp ptr is " << temp_ptr << " and my list is " << my_list << endl;
    //my_list = new_string_ptr;
    //delete [] temp_ptr;
    //new_string_ptr = nullptr;

    delete [] my_list;
    my_list = new string[*list_size];
    my_list = new_string_ptr;
    print_names(my_list, list_size);

Unfortunately the results are the same.