Migrating from ESP8266 to ESP32 web server issues?

I'm trying to resurrect some code and want to migrate from an ESP8266 to an ESP32C3.

In the following routine:

void startWebServer()
{
    // 1. map default requests
    httpServer.onNotFound([]()  
    {   
        if (!handleFileRead(httpServer.uri())) // send file corresponding to URI if it exists in LittleFS
        httpServer.send(404, "text/plain", "404: Not Found"); // otherwise, respond with a 404 (Not Found) error
    });
    
    // 2. map specific requests (with remaping if needed)
    
    // Icon that appears in the tabs of the internet browser
    httpServer.on("/favicon.ico", [](){ handleFileRead("/carre_vert.png"); });

    //Simple mouse shortcut // http Server.on("/sou1", [](){ handle FileRead("/mouse 1.html"); });

    // List the LittleFS content    
    httpServer.on("/list", HTTP_GET, replyFileList2);
    //httpServer.on("/list", HTTP_GET, handleFileList); //try with action in html

    // Upload request
    // if the client posts to the upload page
    // Send status 200 (OK) to tell the client we are ready to receive
    // Receive and save the file
    httpServer.on("/upload.html", HTTP_POST, []() { httpServer.send(200, "text/plain", ""); }, handleFileUpload);
    
    // Delete request (not working) --> workaround: add .del to the file    
    // Edit request (on va pas faire) -->workaround: display the file, save it then load it
    httpServer.on("/edit.html", HTTP_DELETE, handleFileDelete);

    // Get heap status, analog input value and all GPIO statuses in one json call
    httpServer.on("/stat", HTTP_GET, []() 
    {WiFiServer server(80);
        String json = "{";
        json += "\"vers\":" + String(CENTRAL_NAME);
        json += "\"heap\":" + String(ESP.getFreeHeap());
        json += ", \"analog\":" + String(analogRead(A0));
        json += ", \"gpio\":" + String((uint32_t)(((0 | 0) & 0xFFFF) | ((1 & 0x01) << 16)));
        json += ", \"au\":" + String(au);
        json += "}";
        httpServer.send(200, "text/json", json);
        json = String();
    });
    
    // ex with a string in the program (just for testing)
    httpServer.on("/ii", HTTP_GET, []()    // internal index
    {
        httpServer.send(200, "text/html", index_html);  //send the string defined in this file
    });
    
    // Direct AU
    httpServer.on("/au0", HTTP_GET, [](){ au = 0; httpServer.send(200, "text/html", ok_au0);});  //we return not too long so as not to disturb the dcc and not to block the browser
    httpServer.on("/au1", HTTP_GET, [](){ au = 1; httpServer.send(200, "text/html", ok_au1);});

    httpServer.begin(); // start the server
    #if USB_MODE == 1
    DBG.println("HTTP server started.");
    #endif
}

I get compiler errors stating

/Users/etimberl/Documents/Hobbies/Micro Controllers/Arduino/trains/WILDMaterials/WILD_Arduino_Load/D17_32_TinyPico_nospiffs/D17_32_TinyPico_nospiffs.ino: In function 'void no_html_in_pwr()':
D17_32_TinyPico_nospiffs:3495:16: error: 'class WiFiServer' has no member named 'send'; did you mean 'end'?
     httpServer.send(200, "text/html", ask_go_au);
                ^~~~
                end
                                                        ^~~~
                                                              end
/Users/etimberl/Documents/Hobbies/Micro Controllers/Arduino/trains/WILDMaterials/WILD_Arduino_Load/D17_32_TinyPico_nospiffs/D17_32_TinyPico_nospiffs.ino: In function 'void startWebServer()':
D17_32_TinyPico_nospiffs:3763:16: error: 'class WiFiServer' has no member named 'on'
     httpServer.on("/au1", HTTP_GET, [](){ au = 1; httpServer.send(200, "text/html", ok_au1);});
                ^~
/Users/etimberl/Documents/Hobbies/Micro Controllers/Arduino/trains/WILDMaterials/WILD_Arduino_Load/D17_32_TinyPico_nospiffs/D17_32_TinyPico_nospiffs.ino: In lambda function:
D17_32_TinyPico_nospiffs:3763:62: error: 'class WiFiServer' has no member named 'send'; did you mean 'end'?
     httpServer.on("/au1", HTTP_GET, [](){ au = 1; httpServer.send(200, "text/html", ok_au1);});
                                                              ^~~~
                                                              end
/Users/etimberl/Documents/Hobbies/Micro Controllers/Arduino/trains/WILDMaterials/WILD_Arduino_Load/D17_32_TinyPico_nospiffs/D17_32_TinyPico_nospiffs.ino: In function 'void http_loop()':
D17_32_TinyPico_nospiffs:3784:16: error: 'class WiFiServer' has no member named 'handleClient'; did you mean 'hasClient'?
     httpServer.handleClient();
                ^~~~~~~~~~~~
                hasClient
Multiple libraries were found for "WiFi.h"
 Used: /Users/etimberl/Library/Arduino15/packages/esp32/hardware/esp32/2.0.14/libraries/WiFi
 Not used: /Applications/Arduino.app/Contents/Java/libraries/WiFi
 Not used: /Users/etimberl/Documents/Hobbies/Micro Controllers/Arduino/libraries/WiFiEspAT
exit status 1
'class WiFiServer' has no member named 'send'; did you mean 'end'?


This report would have more information with
"Show verbose output during compilation"
option enabled in File -> Preferences.

I replaced the 8266 web library as follows:

//WebServer httpServer(80); // Create a webserver object that listens for HTTP request on port 80
WiFiServer httpServer(80); // Create a webserver object that listens for HTTP request on port 80
WebSocketsServer webSocketServer = WebSocketsServer(81); // Create a webSocketServer object that listens for request on port 81

Should I have found another ESP32 specific library, perhaps?

You don't show all you code (i.e the #include statements) so a bit difficult to tell which library you have actually used, but I am guessing that on the ESP8266 your code probably used:

#include <ESP8266WebServer.h>
ESP8266WebServer server(80);

WebServer actually makes use of and is a wrapper for calls to WiFiServer and the equivalent on the ESP32 is just changing the above to:

#include <WebServer.h>
WebServer server(80);

ESP32 also has an async version which has more functionality and makes use of multi-tasking and multi-processing:

#include "ESPAsyncWebServer.h"`
AsyncWebServer server(80);

As an aside, if you were thinking about going HTTPS, then it gets a bit more complicated because there is no equivalent of ESP8266SecureWebServer. There is an alternative library, but it is not code compatible.