Unable to reinstall OS X on Macbook Air despite trying several different methods I'm out of ideas... Tried every suggestion I could find on the net, but nothing worked. Hoping someone here has a different idea.

The situation 

I'm trying to reinstall OS X (Lion, or higher -- which version doesn't matter to me right now) on a Macbook Air, 11-inch, Late 2010 model. 

The machine came with OS X Lion pre-installed, and has been upgraded up to El Capitan while I used it.

I recently decided to give the MBA away. Before doing so, I erased the default system/home partition with Disk Utility in Recovery Mode.

I have now been trying to re-install OS X on the MBA, but didn't succeed.

Things I tried already


Turning on the machine, i.e. regular startup: Bootup stops at question mark screen. No surprise, since the system partition has been erased.
Turning on with cmd-R, i.e. regular recovery mode. No visible progress for about 30 seconds, i.e. grey screen, then automatically switch to Internet recovery mode. Seems to suggest that there's no recovery partition, or it's not recognized, right?
Turning on with cmd-option-R, i.e. using Internet recovery mode directly. Mac OS X Utilities menus load, no problem here. Selecting option 2, "Reinstall Mac OS X", I am offered to install OS X Lion 10.7 (the original OS the machine was shipped with). Pressing "ok" is followed by message "To download and restore Mac OS X, you computer's eligibility will be verified with Apple". "ok", followed by progress report "Loading Installation Information", then dialog to confirm license agreement, then dialog to select installation disk. After that, I am asked to sign into the App Store. This step is not optional in my case, i.e. I cannot skip the app store sign-in. Once I sign in, installation fails with error message: "This item is temporarily unavailable. Please try again later." 
Checking Disk Utility, I see there's no recovery partition. Not 100% sure why, since I didn't erase it, but given that I got the MBA from someone else as well, it is possible that partition was erased earlier.
Last thing I tried: Recovery Disk Assistant on my working Macbook Pro to create recovery USB. Using USB on MBA at boot up leads to the regular boot up progress bar going about halfway, then stopping at an error screen (box with an "X" inside). Perhaps because the computer I used to create the USB runs El Capitan, while MBA came with Lion.


Summary

Trying to re-install OS X on Macbook Air after erasing default system partition.

Recovery partition missing, so I have to use Internet recovery.

Internet recovery installation fails as follows: 

(1) I am forced to sign into app store when reinstalling OS X Lion (weird, since other people suggest app store sign-in is not necessary when reinstalling original OS). 

(2) After signing in, I get the error "This item is temporarily unavailable."
The best thing you can do is to create a bootable USB disk with any new OS X of your choice. Please see Apple support page to create a bootable disk. I guess, it will solve your installation issues with your MBA.
https://support.apple.com/en-us/HT201372