This is the line of CSS that is being a jerk to you, turning off the bar:
@media(max-width:500px){.pg-header-actions,.pg-header-section-settings,.pg-preset-select-container{display:none}
TamperMonkey script, try on Firefox mobile:
// ==UserScript==
// @name PlaygroundStylerMobile
// @namespace http://tampermonkey.net/
// @version 0.1
// @description Put the control bar back
// @author _j
// @match https://platform.openai.com/playground*
// @grant GM_addStyle
// ==/UserScript==
(function() {
'use strict';
// Inject a style tag into the head of the document
GM_addStyle(`
@media(max-width:500px){.pg-header-actions,.pg-header-section-settings,.pg-preset-select-container{display:block !important}
`);
})();
(CSS Source: https://openaiapi-site.azureedge.net/public-assets/d/bd91eab196/static/css/576.72719439.chunk.css)