On YouTube, open the transcription panel and run this bookmarklet to copy the transcription to your clipboard without timestamps.

bookmarklet.js

javascript: (() => {
    const fragments = [];
    const elems = document.getElementsByClassName("ytd-transcript-segment-renderer");
    if (elems.length === 0) {
        alert("No transcript elements found - are you sure the transcription panel is open?");
        return;
    }

    for (const elem of elems) {
        /* Want timestamps? Remove .slice(1) here: */
        var frags = elem.innerText.split('\n').slice(1);
        fragments.push(...frags);
    }

    navigator.clipboard.writeText(fragments.join('\n')).then(
        function() {},
        function() { alert("Cannot access the clipboard - click anywhere on the page and try again."); }
    );
})();

bookmarklet.min.js

/* Create a new bookmark and set its URL to one of the javascript:... lines below. */

/* No timestamps: */
javascript:(() => {const fragments = [];const elems = document.getElementsByClassName("ytd-transcript-segment-renderer");if (elems.length === 0) {alert("No transcript elements found - are you sure the transcription panel is open?");return;}for (const elem of elems) {var frags = elem.innerText.split('\n').slice(1);fragments.push(...frags);}navigator.clipboard.writeText(fragments.join('\n')).then(function() {},function() { alert("Cannot access the clipboard - click anywhere on the page and try again."); });})();

/* With timestamps: */
javascript:(() => {const fragments = [];const elems = document.getElementsByClassName("ytd-transcript-segment-renderer");if (elems.length === 0) {alert("No transcript elements found - are you sure the transcription panel is open?");return;}for (const elem of elems) {var frags = elem.innerText.split('\n');fragments.push(...frags);}navigator.clipboard.writeText(fragments.join('\n')).then(function() {},function() { alert("Cannot access the clipboard - click anywhere on the page and try again."); });})();