In my case, it turned out that only .zip worked. Neither .mx1 nor .rom ran, however, they all worked well after zipping them. I also tested on Retropie and Batocera, and confirmed they all had the same issue. Could be fMSX's issue. Hope this would help someone in the future.
@paradadf Thanks, but as I mentioned, I added fMSX files on top of vanilla Recalbox. Below is what I have in the bios directory. Do I need anything else? Please let me know!
-rw-r--r-- 1 root root 33152 Jan 4 15:00 CARTS.SHA
-rw-r--r-- 1 root root 2048 Jan 4 15:00 CYRILLIC.FNT
-rw-r--r-- 1 root root 2048 Apr 26 02:08 DEFAULT.FNT
-rw-r--r-- 1 root root 16384 Jan 4 15:00 DISK.ROM
drwxr-xr-x 2 root root 4096 Apr 26 02:31 Databases
-rw-r--r-- 1 root root 65536 Jan 4 15:00 FMPAC.ROM
-rw-r--r-- 1 root root 16384 Jan 4 15:00 FMPAC16.ROM
-rw-r--r-- 1 root root 2048 Jan 4 15:00 ITALIC.FNT
-rw-r--r-- 1 root root 131072 Jan 4 15:00 KANJI.ROM
-rw-r--r-- 1 root root 32768 Jan 4 15:00 MSX.ROM
-rw-r--r-- 1 root root 32768 Jan 4 15:00 MSX2.ROM
-rw-r--r-- 1 root root 16384 Jan 4 15:00 MSX2EXT.ROM
-rw-r--r-- 1 root root 32768 Jan 4 15:00 MSX2P.ROM
-rw-r--r-- 1 root root 16384 Jan 4 15:00 MSX2PEXT.ROM
-rw-r--r-- 1 root root 65536 Jan 4 15:00 MSXDOS2.ROM
drwxr-xr-x 202 root root 12288 Apr 26 02:32 Machines
-rw-r--r-- 1 root root 1009534 Jan 1 1980 NstDatabase.xml
-rw-r--r-- 1 root root 65536 Jan 4 15:00 PAINTER.ROM
-rw-r--r-- 1 root root 16640 Jan 4 15:00 RS232.ROM
drwxr-xr-x 3 root root 4096 Jan 1 1980 amiga
-rw-r--r-- 1 root root 131072 Jan 1 1980 dc_nvmem.bin
drwxr-xr-x 3 root root 4096 Jan 1 1980 fba
drwxr-xr-x 2 root root 4096 Jan 1 1980 keropi
-rwxr-xr-x 1 root root 524288 Jan 1 1980 kick02019.AROS.ext.rom
-rwxr-xr-x 1 root root 524288 Jan 1 1980 kick02019.AROS.rom
-rw-r--r-- 1 root root 5218 Jan 1 1980 lisez-moi.txt
drwxr-xr-x 3 root root 4096 Jan 1 1980 mame2000
drwxr-xr-x 3 root root 4096 Jan 1 1980 mame2003
drwxr-xr-x 4 root root 4096 Jan 1 1980 mame2003-plus
drwxr-xr-x 3 root root 4096 Jan 1 1980 mame2010
drwxr-xr-x 2 root root 4096 Jan 1 1980 np2kai
drwxr-xr-x 2 root root 4096 Jan 1 1980 oricutron
-rw-r--r-- 1 root root 5104 Jan 1 1980 readme.txt
@paradadf Thanks so much for your help. Happy I'm not yet abandoned alone 
Actually, I have already tried everything I can. Of course I tried with blueMSX and fMSX with different versions.
One thing I found meanwhile, there is an error when I run a game in verbose mode via ssh:
[INFO] [MIDI]: Initializing ...
ALSA lib seq_hw.c:466:(snd_seq_hw_open) open /dev/snd/seq failed: No such file or directory
[ERROR] [MIDI]: snd_seq_open failed with error -2.
[ERROR] [MIDI]: Initialization failed (list of input devices unavailable).
Should this be a reason? This is a vanilla install with fMSX BIOS only. If you want me to try something different for logging, please let me know. Thanks!!
@Storm Have you been able to fix this issue? I am experiencing a quite similar issue, even on the latest Recalbox 6. Can you tell me how you resolved it? I'm on Rsbp 3+. Thanks!
@novaflash Have you been able to fix this issue? I am experiencing the same issue, even the latest Recalbox 6. Can you tell me how you resolved it? Thanks!