I'm gonna go against the grain here and recommend a Swap Magic disc coupled with a USB flash drive that you will consider "the PS2's flash drive", as without it you won't be able to run any homebrew. Put a copy of uLaunchELF on the USB drive as SWAPMAGIC/SWAPMAGIC.ELF, and when booting SM with this USB drive inserted, it will automatically launch whatever ELF you put in that location, which is why I recommend uLE. Once you get uLE running, you can run any other homebrew, and in fact you can even launch the hacked OSDSYS that FMCB provides, from a working FMCB installation on a MC (in fact, you can even have installed it with that same PS2, that can't actually boot FMCB). This was dlanor's recommended method of running homebrew on a newer PS2 incompatible with FMCB, and since we don't see him much anymore, I'll continue his suggestion on this.
You won't even need the "magic switches" or whatever that allow you to swap discs if you do it this way, all you'll need is an original SM disc (buy the CD and DVD set, and you'll have a backup that you can store away for a rainy day should something happen to whichever disc you choose as your primary boot method). Much less hassle than sourcing a good modchip and installing it. The only major thing that the modchip can really do that SM+uLE+FMCB can't do is boot PS1 backups.
Chip-free/MC-free homebrew <3
FMCB 1.93 on MC for emergencies, NA
250GB HDD/FHDB 1.93/OPL 0.9.2 (GSM and non-GSM)/PS2 games, NO HDDOSD
80GB HDD/FHDB 1.93/POPStarter rev12/OPL644/GSM0.37/PS1 games, NO HDDOSD
SCPH-75001 (sitting idle, slims suck):
FMCB 1.92 on MC