I finally got something working, maybe not the most awesome-looking code, but it works perfectly. I'll put my code in case people wanna try this.

The backside of this is that it will only work with a site that has the same components:
- A custom folder as parent for all page that needs the breadcrumb (including all parents' folders). Only pages right after the language node doesn't need a folder.
- A sub-page as the index of the folder
- A language node as first child node of the site (ex: "mysite.com/en/...")

Of course, this code can be customized to fit what ever needs you have.

To start, I created a new custom Folder called "Breadcrumb Folder" with 2 properties:
- FolderDisplayName (to be displayed on breadcrumb link)
- FolderMainPage (main page of the folder (since the folder is not a valid page))

Then I created a new webpart that links to a usercontrol I created as follow:

breadcrumb.ascx.cs

protected void Page_Load(object sender, EventArgs e)
    {
        // The breadcrumb HTML container
        StringBuilder breadcrumb = new StringBuilder();

        // Get site root (custom method to get "http://mysite.com")
        string root = SiteUtils.GetSiteRootUrl();
        root.Trim();
        breadcrumb.Append("<ul id=\"breadcrumb\">");

        // Create the first element (main page) with a ressource string
        breadcrumb.Append("<li class=\"first\"><a href=\"" + root + "\">" + SiteUtils.GetString("accueil.text") + "</a></li>");

        // Get relative URL of current document
        string url = CMSContext.CurrentDocument.DocumentUrlPath;

        // Remove first slash and page name with last slash
        url = url.Substring(1, url.LastIndexOf('/') - 1);

        // Split into folders
        string[] folders = url.Split('/');

        // Get folders' count
        int foldersCount = folders.Length;
        // Track the amount of folders that can be viewed in a loop
        int currentFolder = foldersCount - 1;
        // Keep track of the loops count remaining
        int folder = foldersCount - 1;

        // Build the Breadcrumb (excluding the language)
        string currentURL = string.Empty;
        for (int z = 0; z < foldersCount; z++)
        {
            // Set amount of folders remaining to loops through
            currentFolder = folder;

            // Set current url (ex: "/my-folder-name/second-folder")
            currentURL += "/" + folders[z];

            // Init the vars that will contain the folder properties
            string folderDisplayName = string.Empty;
            string folderMainPage = string.Empty;

            // If it is not the first level of child (the language node)
            if (z != 0)
            {
                //Get the current node
                CMS.TreeEngine.TreeProvider tp = new TreeProvider();
                CMS.TreeEngine.TreeNode currentNode = tp.SelectSingleNode(CMS.CMSHelper.CMSContext.CurrentSiteName, CMS.CMSHelper.CMSContext.CurrentAliasPath, CMS.CMSHelper.CMSContext.CurrentPageInfo.DocumentCulture);

                // As long as the node is not empty, loop through the parent
                while (currentNode.NodeParentID != null)
                {
                    // If there is still a folder and a loop remaining
                    if (currentFolder > 0 && folder > 0)
                    {
                        // Get the parent node and reduce folder count
                        currentNode = tp.SelectSingleNode(currentNode.NodeParentID);
                        // Reduce the folder count
                        currentFolder--;
                    }
                    // Otherwise, set the vars with the properties of the current folder obtained with loop
                    else
                    {
                        folderDisplayName = (string)currentNode.GetValue("FolderDisplayName");
                        folderMainPage = currentURL + "/" + (string)currentNode.GetValue("FolderMainPage");

                        // Reduce loop count
                        folder--;
                        break;
                    }
                }
                // Add the page to the breadcrumb if it is not the main page of the folder
                if (folderMainPage != CMSContext.CurrentDocument.DocumentUrlPath)
                    breadcrumb.Append("<li><a href=\"" + root + folderMainPage + ".aspx" + "\">" + folderDisplayName + "</a></li>");
            }
        }

        // Add the current page title to the end of the breadcrumb
        breadcrumb.Append("<li class=\"last\">" + CMSContext.CurrentDocument.GetValue("DocumentPageTitle") + "</li>");
        breadcrumb.Append("</ul>");
        ltlBreadcrumb.Text = breadcrumb.ToString();
    }

breadcrumb.ascx

<%@ Control Language="C#" AutoEventWireup="true" CodeFile="breadcrumb.ascx.cs" Inherits="Controls_common_breadcrumb" %>
<asp:Literal ID="ltlBreadcrumb" runat="server" />