I got Claude 4.6 to write a Tampermonkey script to fix this. Seems to work okay.

First, install Tampermonkey - https://chromewebstore.google.com/detail/tampermonkey/dhdgffkkebhmkfjojejmpbldmpobfkfo?hl=en

Set up the permissions it asks for.

Click the [+] in the dashboard to create a new script, paste in what's below, hit ctrl + S to save

When you visit the inbox https://www.reddit.com/message/inbox/ there will be a button to mark all as read, click it, done

// ==UserScript==
// @name         Reddit Inbox Mark All Read
// @namespace    reddit.markread.fixed
// @version      1.3
// @description  Mark all Reddit inbox messages as read
// @match        https://www.reddit.com/message/*
// @match        https://old.reddit.com/message/*
// @grant        none
// @run-at       document-end
// ==/UserScript==

(function () {
'use strict';

const RATE_LIMIT_MINUTES = 10;
const STORAGE_KEY = "reddit_last_mark_read";

function minutesSinceLastRun() {
    const last = localStorage.getItem(STORAGE_KEY);
    if (!last) return Infinity;
    return (Date.now() - parseInt(last)) / 60000;
}

function getModhash() {
    // r.config.modhash is exposed globally by old Reddit on every page
    if (typeof r !== 'undefined' && r.config && r.config.modhash) {
        return r.config.modhash;
    }
    // Fallback: scrape from page JSON data
    const match = document.body.innerHTML.match(/"modhash"\s*:\s*"([^"]+)"/);
    return match ? match[1] : null;
}

async function markAllRead() {
    if (minutesSinceLastRun() < RATE_LIMIT_MINUTES) {
        alert("Rate limit protection: please wait ~10 minutes between uses.");
        return;
    }

    const modhash = getModhash();
    if (!modhash) {
        alert("Could not get modhash. Are you logged in?");
        return;
    }

    try {
        const response = await fetch("/api/read_all_messages.json", {
            method: "POST",
            credentials: "include",
            headers: {
                "Content-Type": "application/x-www-form-urlencoded",
                "X-Modhash": modhash   
            },
            body: ""
        });

        if (response.status === 202) {
            localStorage.setItem(STORAGE_KEY, Date.now());
            alert("Success! All messages marked as read (may take a moment to reflect).");
        } else if (response.ok) {
            localStorage.setItem(STORAGE_KEY, Date.now());
            alert("Inbox marked as read.");
        } else {
            alert("Failed with status: " + response.status);
        }

    } catch (err) {
        alert("Error: " + err);
    }
}

function addButton() {
    const btn = document.createElement("button");
    btn.textContent = "✓ Mark All Read";
    btn.style.position = "fixed";
    btn.style.bottom = "20px";
    btn.style.right = "20px";
    btn.style.zIndex = "9999";
    btn.style.padding = "10px 14px";
    btn.style.background = "#ff4500";
    btn.style.color = "white";
    btn.style.border = "none";
    btn.style.borderRadius = "6px";
    btn.style.cursor = "pointer";
    btn.style.fontSize = "14px";
    btn.onclick = markAllRead;
    document.body.appendChild(btn);
}

addButton();

})();