Windows Error Reporting service is by default set to "Manual (Trigger start)" and currently was stopped.
But it seems I have found the cause of the issue. I also have seen some other threads where people were having some minor issues with Intel management engine interface on Windows 7, but it seems, what is just a minor issue on older Windowses, becomes a catastrophe on Windows 10.
Essentially, what I did was enabling boot logging mode and then letting it try to unsuccessfully boot for some minutes. Then I rebooted from my primary OS and was really glad to see that ntbtlog.txt was created on my Windows 10 partition because I read somewhere that it might be missing if Windows gets stuck before Session Manager starts.
So, at the end of ntbtlog.txt I see lots of lines
There's nothing else between them, only these lines.BOOTLOG_LOADED \SystemRoot\System32\drivers\TeeDriverW8x64.sys
Oh, Microsoft, why no timestamps for events? Anyway, I started thinking that maybe these lines matched those short bursts of LED activity which I saw while PC was trying to boot. Most probably, it tried to load TeeDriverW8x64.sys and wrote that to the log (hence the HDD LED flash), then maybe failed (but why no any failure messages in boot log?) and then decided to repeat it forever.
While looking for TeeDriverW8x64.sys on the Internet it led me to Intel(R) Management Engine Interface (IMEI) driver. So while I was on Win7, I looked in Device Manager, and IMEI was marked yellow with code 10. I tried to install the newset drivers, from Intel, but installer reported that the current drivers are even newer - so, Intel's site has out-of-date IMEI drivers. Then I stumbled upon a forum post which described that part of IMEI is actually embedded into BIOS firmware. Oh, I can reset BIOS! So I shut down the PC, removed CMOS battery, rebooted and for a short moment saw a cryptic message from BIOS which I haven't seen any other times when booting my PC:
It was counting percents and when it reached 100%, it booted to Windows 7. And hooray, no more warnings in device manager, and also event viewer did not have those messages.Fix ME Firmware Data
And even better - Windows 10 also booted just fine.
So, I've been living with some IMEI issue for years on Windows 7 without any issues, but Windows 10 made a big deal about it.
Now when I know what to look for, I have found also similar stories from other Windows 10 users who have suffered from BSODs related to IMEI and they had to completely disable IMEI in device manager, if they are lucky enough to be able to boot into Windows.
I hope my experience will help other people to deal with IMEI. Definitely it is one of the first things worth checking before upgrading to Windows 10 and after the upgrade in case of BSODs.
BTW, ntbtlog.txt starts with some weird symbols
ªÝàÿÿft (R) W11 2 2015 20:33:03.350
but I hope it does not mean that something has gone wrong... anyway, I'll reinstall Windows 10 from scratch, now that I hopefully know that it works.