You can with this,

html("
    <style>
        .hud-menu-button.i-32-24.i-white.i-setting-tool.--ul-admin-button.admin-button{display:none;}
    </style>
")

but it will only work when the Form where the Formula field is is open. You can put this in Trigger after open,

dialog("Hide Wrench", html("<script>
    var styleSheet = document.styleSheets[2];
    styleSheet.insertRule('.hud-menu-button.i-32-24.i-white.i-setting-tool.--ul-admin-button.admin-button{display:none;}');
    var interval = setInterval(() => {
        var bt = document.querySelector('.nx-alert .nx-button-text');
        if (bt) {
            bt.click();
            clearInterval(interval);
        }
    });
</script>"), ["close"])

but then you'll either have to edit the database file manually to to make Admin button display again or set up a way to show the Admin button in a Formula field which kind of defeats the purpose unless you hide it.

If you use the Trigger after open version you can put this in a Formula field and if it isn't hidden and the Form is open it will show the wrench.

html("
    <style>
        .hud-menu-button.i-32-24.i-white.i-setting-tool.--ul-admin-button.admin-button{display:inline;}
    </style>
")