MacBook Air 2015; Login and recovery mode failed
Can you provide screenshots/videos of the console text during verbose boot?
Do you have a second computer and a USB or Thunderbolt cable that you can use for booting your MacBook into Target Disk Mode (hold T during boot) to run Disk Utility on the hard drive?
(Unlikely to help but might be worth a shot if convenient.) Do you have an external display that you can attach to the MacBook and try booting while in clamshell mode? (Attach the display with the computer off, press the power key, and close the lid.) Trying to wiggle a GPU-related variable here, given your IGPU timeout messages.

Ultimately you may need a reinstall of macOS but let’s try to debug incrementally. The fact that single user mode works is a very important clue.
UPDATE 1:
With the additional data,
I can’t tell immediately from your boot logs what the root cause is, but it’s clear that your IGPU driver is unhappy. Single user mode works because it doesn’t require switching to accelerated graphics (but, then, neither should Safe Mode).
It may be interesting to capture logs from the point after you enter single user mode and then type exit. Make sure you’re entering verbose mode first (i.e., hold CommandV upon reboot and then immediately switch to CommandS once you see console text). I suspect we’ll stall at IGPU timeouts again.
Beyond this point, you can try two paths.

We can wait until you get a second computer available for Target Disk Mode and try to debug this incrementally.
We can try Internet Recovery and try either a newer or older OS via OptionCommandR or ShiftOptionCommandR, respectively. (How to Install macOS via Recovery Mode)

I would personally give #2 a try right now.
UPDATE 2:
It’s unusual for Recovery mode to fail to transition to graphics mode while single user mode works fine. The new logs didn’t reveal a root cause to me either, other than reconfirming that we aren’t able to enable accelerated graphics in the GPU and thus enter graphics mode.

I’m always reluctant to suspect a hardware failure until we’ve ruled out software issues, which are far more common, but I would encourage you to run Apple Diagnostics if you haven’t already: Hold D during boot.
If diags passes, use Target Disk Mode to run Disk Utility on your SSD through your second computer.
If the scan completes successfully and finds no corruption, I would take this opportunity to make a full backup of your drive, as it’s looking increasingly likely that you’ll need to do an erase-install as a workaround in lieu of root-causing this yourself.