Firmware updates do not remove BC. However, old fully BC PS3 models require the following (actually probably true of most PS3 models, but especially earlier variants):
1. Delidding of CELL and RSX and reapplication of thermal paste. My understanding is that Sony used a mediocre thermal paste which typically would quickly dry out and fail to conduct heat away from the chips, or come out of contact with the integrated heat spreaders attached to the chips. Both IHS are attached with an extremely adhesive silicone kind of glue, and you will need to very carefully cut this away to remove the IHS, without damaging CPU traces, or snapping off other components. Look up a a repair video.
2. Once this is done, these models typically run very hot as Sony seemed to prioritize quietness over thermals. To fix this an ensure longer term functionality of the unit you will most likely need to install some type of custom firmware and fan curve modifier, to keep the unit at lower temperatures. If your unit has thermal paste in good nick (i.e. if it is comparatively not used), this alone might be sufficient to get decent working life out of it. PS3 CFW comes with its whole set of issues, and will effectively make your unit an offline machine only as Sony is extremely aggressive about banning accounts that go online with CFW PS3 devices. There are some ways around this, none seem entirely foolproof and there's not a whole lot of insight into the telemetry.
2a. Custom firmware is also essential because Sony decided the sensible thing to do was to marry a bunch of components, most notably the Blu Ray drive to the unit's motherboard, making swaps impossible. Additionally, if a drive is not connected to your device on OFW it might not boot. CFW gets around this by removing the BD flag, and letting you mix and match blu ray drives (with a lot of additional conditions, but still). Essential for long term repairability and diagnosis of the system.
2b. Custom firmware also unlocks region for all content, and will let you use Sony's software emulator (ps2_netemu) for PS2 classics, for all your library. You will need to dump discs to .iso. Compatibility is extremely decent, despite lacking the hardware. I think last I checked something like 80% of PS2 titles either played fine, with minor issues, or had PS2 classics officially --- this way you might be able to use any old more modern PS3 with less issues likely to crop up to play your games. Check a compatibility list.
3. NEC/Tokin Capacitors. These will probably fail at some point. They were notoriously bad caps that were used because of a quite high capacitance in a slim form factor. The replacement isn't impossible by any means, but requires removing the old caps from the PCB which can be an involved process, requiring removal of the old ones and soldering. These were replaced in slim and super slim models. Slims after around 2010/2011ish I think, but check a list of model and rev numbers.
4. Fan replacement and power supply replacement -- there are larger fans I believe you can swap in to help thermals, as well as the launch FATs can be upgraded to a more power efficient and less hot power supply.
Personal recommendation: Retroarch and a PS2 core. Owning and continuing to operate an old PS3 requires a lot of additional baggage you need to be willing to deal with, and emulation will let your games look very good.
