Long fsck instead of GRUB menu after fresh install of stable EOS 6.1 Jolnir

### What Happened?

Every boot of my laptop lasts about 1min. The system is installed on an SSD and the EOS 5.* Hera was booting up in a few seconds. Here in OS 6.1 I have a cursor at the beginning, then after ~30sec I see a `/dev/sda2: clean, */* files, */* blocks` message(seems like it's `fsck`) and then goes the rest of the boot log(Plymouth doesn't work ATM, it's another issue).
It's a dual-boot setup: EOS 6.1(previously EOS 5.1.7) + Windows 10 on a single SSD. Also a 1TB HDD is installed, but removing it from the laptop doesn't solve the issue at all.
This happens on **every** boot, the shutdown process every time is normal with no troubles.
Also I don't see the GRUB menu at all. Seems like it's not working at all, because I tried `boot-repair`'s automatic repair(always fails in the end) and changing GRUB config by myself using the GRUB Customizer(tried to change the timeout and added an option `fsck.mode=skip` to the `GRUB_CMDLINE_LINUX_DEFAULT`) - no effect.
Maybe there's some EFI issue?

Here's a video: https://youtu.be/5CLIbXU5IRI

### Steps to Reproduce

1. Installed EOS 6.1 stable from a bootable USB.
2. Reboot after installation and every subsequent power up/reboot.

### Expected Behavior

Skip `fsck` as there are no problem, show the GRUB menu.

### OS Version

6.x (Odin)

### Software Version

Latest release (I have run all updates)

### Log Output

```shell
No log as it happens before logs are being written to the disk(or I don't know where to get such logs).
```

### Hardware Info

Lenovo Y580, the OS is installed to an SSD.