Under gold standard, the gold content of a currency is determined by the government and can be adjusted at any time. The currency can still be devalued by the government. A gold-based currency is better than fiat, which is based on nothing. If Fed wants a massive QE under gold standard, it has to devalue the dollar first, which makes QE look bad. That's why there is no QE under gold standard. It shows the financial credibility of a country.
The first two sentences in your post seem correct. Most of what follows those sentences, however, does not. QE does not cause inflation. It is used when demand for money is too low. QE increases bank reserves putting downward pressure on the wholesale price of money either via the old way of controlling the funds rate indirectly, by controlling aggregate bank reserves, or by the new method. Nowadays, the Fed can still use QE if it believes bank reserves are too low, but it also changed the method it uses to control the funds rate. When the Fed wants the wholesale price of money to drop it pays lower interest on reserves, putting a lower floor under the funds rate. It adjusts the discount rate to establish an upper bound for the funds rate. No bank will lend to another at a rate below the risk free rate of return it gets from the fed nor will it pay more for money (usually) than it can borrow via the discount "window". (The funds rate is the wholesale price that banks have to pay for the money they lend.)
You wrote
"Fiat [is] worthless" as though you were referring to Bitcoin, but even Bitcoin is not worthless so long as someone wants it. Money, including fiat money, and gold too, is worth what it can be exchanged for, but of course gold also has intrinsic value.
Our Fiat money is directly backed by our government's taxing power --- which is considerable. There will be created enough demand for fiat money as needed to pay taxes, because fiat money is the only thing our government accepts in payment. Beyond that, it's the productivity of nations producing goods and services for sale in U.S. Dollars that backs fiat dollars. More goods services and assets are for sale in U.S. Dollars than are for sale in any other currency. That's quite a strong backing!!!
You also wrote,
" If Fed wants a massive QE under gold standard, it has to devalue the dollar first, which makes QE look bad. That's why there is no QE under gold standard."
You may be unwittingly and indirectly referring to how horribly cumbersome a gold standard is. But you are not correct when you write,
"If Fed wants a massive QE, it has to devalue the dollar..." Your misunderstanding comes from belief that QE involves increasing the amount of money in the private sector. Actually QE leaves the total money in the private sector unchanged. Instead, it simply changes one form of our money into another form without changing the amount.
U.S. Treasuries are just another form of money --- albeit an interest paying form that serves as a store of money. U.S. bonds are still money however, and therefore they are freely convertible to and from bank reserves. In QE all that happens is that X dollars in the form of Treasury securities disappears and is replaced by X dollars in the form of Bank reserves. The total amount of money in the private sector economy is unchanged!
What used to happen when the Government wanted to increase the total amount of money in the private sector without changing the underlying value of its money relative to gold? Under a gold standard the government had to buy more gold. Therefore the government would create,
"out of thin air," "gold certificates" which are both legal tender and convertible at the Treasury to gold! It used those certificates it just printed to buy gold. Then the seller of the gold caused the certificates to circulate in the economy when it used the certificates as legal tender. That's right!, the government created gold certificates "out of thin air" and used them to buy gold!!!
Now do you see why Gold Standard money is really not fundamentally different from Fiat money, but just ridiculously more cumbersome. A gold standard is so cumbersome that Franklin Roosevelt could not wait for new gold certificates to be printed, used to buy gold, allowed to circulate as legal tender, and then perhaps be converted to non-gold-convertible federal bank notes later. He needed to expand the money supply in a hurry as the Nation was still in the grips of a depression. The quickest way to do that was to devalue the dollar by printing and spending more dollars into the economy without bothering first to acquire more gold. The government did call in privately held gold. Roosevelt had massive unemployment on his hands. He had to create many new jobs, put the nation back to work, and get money circulating once again. Technically he did not take us off the gold standard but simply engaged in a massive devaluation, which the furious central bankers nevertheless characterized as taking us of the standard. . It worked just as Keynes told Roosevelt it would.
Fiat money makes infinitely more sense than a money based on a commodity. The latter requires that the issuer of commodity based money be able to control the market for the commodity into which their money is convertible. We will never go back to commodity base money. When J.P. Morgan said "Only gold is money", he was wrong. But how could he know better?, because the World thought as he did for centuries.
Some people believe Noah loaded one of every animal species on earth onto his Ark, which is impossible. And some people will, I suppose, always think we should go back to a gold standard, despite that being impossible too. . .