When the Federal Reserve uses quantitative easing, it adds more money to banking ledgers and removes investment opportunities. The amount that they can do is limited by the reality of the economy. Too much and they cause inflation. Not enough and there is deflation. While nominally under the control of the Fed, in reality the amount is determined by the situation.
Note that during recessions, banks tend to increase their reserves, which somewhat counteracts the effect of quantitative easing.
Mortgage-backed securities vs. government bonds
The biggest issue here is that they purchased mortgage-backed securities instead of the more traditional treasury securities. By doing so, they kept mortgage-backed security prices artificially high and saved investors from their bad decisions.
Treasury securities are government debt. So if they had stuck with buying that, as they normally did during crises, that would have kept the cost of borrowing by the government artificially low. This particular crisis was exceptional not in using quantitative easing but in using it to purchase mortgage-backed securities rather than longer term versions of the short term government debt that they normally buy.
Why subsidize mortgage-backed securities? Barack Obama made a deliberate effort to return the housing and mortgage markets to "normalcy" as quickly as possible. Mortgage-based securities are owned by financiers who support the mortgage market and thus the housing market. So bailing them out had an indirect impact on the housing market.
A cynical interpretation would be that bankers and other financiers make political donations and enjoyed the results as bailouts. Less cynically, there is a belief that if financial institutions fail, there will be negative side effects.
This was somewhat successful. In many areas, house prices have already recovered to their prerecession values. Of course, it is unclear if that means that they are recovered to their natural levels or inflated back into a bubble.
Why is inflation a worry now but not then? During a recession, banks don't see a lot of good loan opportunities. As a result, they loan less. However, with a reserve requirement of 10% (the value in the US), 90% of the money in the economy is created by bank loans. If they make fewer loans, that means less money in the economy. Federal Reserve purchases of debt (short or long term government bonds plus mortgage-based securities this time) swell the cash portion of bank reserves. This encourages banks to find places where they can invest it.
During the recession, the primary worry was that there was too little money in the economy, which can cause deflation. Deflation has the same risk problems as inflation plus it causes employment losses. So we'd rather see inflation than deflation. So they were looking to expand the money supply. Quantitative easing was the method.
After the recession, that shifts. Banks start lending again, creating money on their own. So the Fed wants to soak up most of the money that they created during the recession. They do this by selling the mortgage-backed securities and government debt that they bought during the recession.
The debt ceiling
Another issue is that the debt ceiling includes debt owed to the Fed. So this wouldn't actually help the particular problem of needing to raise the debt ceiling. Whether the bonds are owned by the Fed, banks, government trust funds (e.g. Social Security), private individuals, or whatever, they all count against the debt ceiling.
TL;DR: So the literal answer to your question is that it wouldn't work as Fed reserves count against the debt limit.
To bailout the government like they did mortgage securities, the government would have to issue a large coin (say $1 trillion) and deposit it with the Fed. The Fed would then send back cash in return. The government would spend the cash like any other money they receive. That would not count against the debt limit because the government can coin money. It would still cause inflation, as it expands the money supply. So it would replace other inflation causing activity.