Q:

Could a VWP overstay due to a pending asylum application result in being denied entry to the US?

I understand that if someone previously overstayed a visa waiver (ESTA) by 6 years, that person should be banned for 10 years.
Now to be banned, you need to accumulate more then 365 days of overstay.
Now the law says that you DO NOT start accumulating days if you have a pending case with the immigration offices for asylum.
Now if after 6 years of pending asylum case you are being denied asylum (and ordered removed), and you leave voluntarily, will you be banned from re-entering?
More details: 

I entered originally in 2011 (through LAX, on ESTA).
Applied for asylum in 2012 (within 1 year of entering).
2017 denied asylum, ordered removed.
Left US in 2017 right away voluntarily.
Now having same original passport, just applied for new ESTA, was approved.

Also, we applied WITHIN the 90 days limit, so we never stayed there NOT even a single day illegally. 
Question:
In this case, will I be banned from re-entering via LAX (on VWP)?

A:

You're probably not formally banned, but that just means that a border guard will be allowed to let you in without losing his job over it, if he believes you're really a genuine tourist or business traveler.
But they never have to let anyone in. And with the background you're presenting here you will have a devil of a time convincing any border guard that this time you really truly actually intend only to visit. Then you'll be sent back anyway.
Also, if your asylum application was filed after your 90-day VWP status ran out, then you have overstayed on VWP and even one day of VWP overstay means that you're forever barred from entering under the VWP again. It would be helpful if the ESTA system detected this and rejected your application outright, but this doesn't mean that you won't be turned away when actually arriving in the US, if they discover the overstay at that point in time.
All in all, this sounds like a lost cause.

A:

To add to @Henning Makholm answer, your case is problematic in more ways than one and you're almost certainly guaranteed to be denied on more grounds than one.
Firstly you have demonstrated immigrant intent by your claim for asylum the first time around. That is a disqualifier for ESTA and non immigrant visas. At this point considering you fought this case for six years, nothing shows your intentions have changed and a reasonable immigration or consular officer (if you should apply for a visa) will deny your application.
Second you may even have committed immigration fraud and a zealous immigration officer could say you entered the previous time on ESTA knowing full well you were going to claim asylum. That's Preconceived intent, and that is basis for immigration fraud and that carries a permanent ban.
The fact that you did not accumulate more than one year or even one day of illegal presence (per your timeline) just means legally you do not have a bar and that you are not inadmissible. In practice however almost no consular officer or airport/border immigration officer will let you into the country knowing your immigration history. I mean it doesn't make sense for them to.
So even if you are able to argue that you never overstayed your ESTA because you filed asylum before your ESTA expired, your overall history makes you very very very unlikely to ever enter the USA on a non-immigrant visa.
PS: Even if you left the USA just one day after your asylum petition was denied, it means you had started accumulating unlawful presence from the day of that ruling unless you were specifically granted voluntary departure.

