fsck before mounting at boot is not “a convenient way to do it”. It is done so for ages. It is part of the boot sequence.
Because the system is not as stupid programmed as you seem to think. The general rule is of course: eevry fs that should be mounted at boot, because /etc/fstab says so, is first fsck’ed. So no entry for /home, then no fsck and no mount.
No, it is not. GRUB reads the kernel from the partition because it is told so in the boot configuration. Grub knows enough of the file system type used (and that explains why not all file system types are fit for this and sometimes people have to revert to a separate /boot partition) to find the kernel.
Somewhat later (when the kernel has the appropriate fs dribers loaded) the root file system is mounted read-only to be able to load a lot of stuff. Because of it being mounted read-only, an fsck can be done (and is done). Then the root file system will be re-mounted writable.
This is all nice to know (and it works like that for about 30 - 40 years), but it is far off topic IMHO.
The topic being (again IMHO) that you, for some reason, try to fsck a file system. You did it wrong and we explained you how to do it correct.
About your quest to meddle around with the fsck part of the boot process, my advice is not to change anything there before you have an in depth knowledge about the boot process, and the internals of file systems and what can go wrong there.
And this is from my /var/log/boot…log:
fsck from util-linux 2.23.2
[/sbin/fsck.ext4 (1) -- /] fsck.ext4 -a /dev/sda2
System_A: clean, 168117/1313280 files, 1483970/5242880 blocks
fsck succeeded. Mounting root device read-write.
Mounting root /dev/root
mount -o rw,acl,user_xattr -t ext4 /dev/root /root
And this is later on when it comes to mounting my other file systems:
systemd-fsck[420]: Home_A: clean, 57156/6094848 files, 13768466/24352531 blocks
^[32m OK ^[[0m] Activated swap /dev/disk/by-id/ata-Hitachi_HDT725032VLA380_VFJ201R23XUEXW-part1.
^[32m OK ^[[0m] Reached target Swap.
^[32m OK ^[[0m] Started File System Check on /dev/disk/by-id/ata-Hitachi_HDT725032VLA380_VFJ201R23XUEXW-part3.
Mounting /home...
^[32m OK ^[[0m] Found device Hitachi_HDT725032VLA380.
Starting File System Check on /dev/disk/by-id/ata-Hitachi_HDT725032VLA380_VFJ201R23XUEXW-part6...
^[32m OK ^[[0m] Mounted /home.
^[32m OK ^[[0m] Found device Hitachi_HDT725032VLA380.
Starting File System Check on /dev/disk/by-id/ata-Hitachi_HDT725032VLA380_VFJ201R23XUEXW-part5...
systemd-fsck[426]: Home_B: clean, 55796/6553600 files, 18259908/26214055 blocks
^[32m OK ^[[0m] Started File System Check on /dev/disk/by-id/ata-Hitachi_HDT725032VLA380_VFJ201R23XUEXW-part6.
systemd-fsck[433]: System_B: clean, 163320/1313280 files, 1487704/5242880 blocks
I can assure yoou that any warnings and/or arrors and/or error reporst will be thre when applicable. Thus when want to inspect that after a boot, you are free to do so.