I have watched numerous videos of people installing Hollowtech bottom brackets after removing the square taper ones. I have yet to see them being faced. Is facing really important?

It might be important. Depends on the surface condition of the bottom bracket shell.
I guess it also depends on whether or not the shop has a 'facing tool'. Likely many today don't.

And those that do just as likely have no idea how to use them.

I had a badly tweaked custom frame chased once by a frame builder years ago who straightened the frame and I think he added only $25 for the BB chasing which he said probably was not needed as the custom builder would have done it and I had no reason to doubt condition and the threads were good. If you are moving to outboard bearings and saw lots of paint or other reason maybe but unless your heading out on a tour a shortened bearing life might not be a big deal and could then have it done.

important ? required ?

depends

but the bottom bracket shell and head tube should be faced

( regardless of crank / bb type - unless special case … press fit or something else different ?)

Most decent bikes today won't need to have their BB shells faced. Many decent bikes BITD didn't need it either. If you bought a vintage bike from the past that was actually one of the cheap bikes made in the past but now for some reason are highly regarded, you might need to face the BB if it hasn't already been faced. Like back then, if you are buying a cheap BSO today, it might need it's BB shell faced.

The differences in type of bearings used in the BB today also make them more forgiving of small deviations that some in the past wanted to be persnickety about.

Facing and chasing a BB are two different things. As far as facing being required, what is required is to figure out if the faces are square and parallel, there are a few ways to do that, and most times they are GTG on most frames.

Typically a BB needs to be faced once, when new. Obviously if there is paint or damage to the faced surface, it should be done again but otherwise, no need.

If you are purchasing a brand new frame which hasn't had any components installed then yes, you will need the bottom bracket faced and chased. If you are just replacing a bottom bracket on a bike which already had the bottom bracket installed then you don't need to face or chase the bottom bracket shell unless it's damaged and isn't flat or the threads are damaged. It's very important that the bottom bracket shell is prepared correctly, you can't just install a bottom bracket on a shell that's not flat or has threads messed up.

I assume manufacturers of carbon frames now rely on decent manufacturing tolerances but does anyone know for sure? For those who see bare frames, is there any evidence of facing from the factory? Or maybe that's part of the cleanup between mold and paint and that's good enough?

Might, might, might, possibly....

It's pretty simple. If your bottom bracket hardware contacts the face, it needs to be even (faced). Otherwise the fixed cup (or the lockring) will not have contact evenly, all around the face. This can make the spindle line up crooked on the other side, even. If your BB hardware never contacts the face, but only the threads, facing is not needed. Phil Wood BBs, and the NDS of Shimano cartridge BBs do not depend on a faced surface. Dunno about Hollowtech...

Facing should only need to be done once when the bike is first manufactured and does not apply to carbon frames.

In some cases with steel frames "touch up" facing may need to be done if original was not done cleanly or for other corner cases

Having assembled bikes at a shop that faces every bb and head tube on the new bikes they sell, my experience is that almost every bike, up to the most expensive, need to be faced. This is particularly true of aluminum bikes, which often have significant distortion back by the chainstays. There are lots of bikes running around with minor bearing misalignments due to this. The good news is that threaded cups will average out the distortion. Doesn't mean that it isn't having an effect.

Since I first read this and after my response I've been thinking about it off and on. I'm 68 and retired...not a lot to do in the winter lol and my Zwift race doesn't start for 2 more hours.
When I worked in a bike shop back in the mid '80's to early 2004 I seem to remember not having to 'face' many bottom bracket shells. If I remember correctly all the bottom brackets were internal bearings...in a cage or the better ones, loose bearings...and the race shells were near flush with the outboard face of the bottom bracket shell on the adjustable side. The fixed side would get faced. Generally this was done on steel frames...which most were...because of the paint and top coat. This needed to be removed for a flush/flat fit of the fixed bearing cup. Same on any frame that was painted. We always faced the head tube. We always chased the bottom bracket threads just in case there was overspray. You certainly don't want to cross thread a bottom bracket...yikes it is not pretty but it can be saved with patience, effort and prayers/offerings to the bike gods. We generally chased all threads just because it was the right thing to do when the bike is being built and a lot easier to do then than later...which may mean a cross thread problem.

Most bike shops build far more "bikes in a box" than "frame in a box" during the year. They also come nearly complete so the bottom bracket and head set is already installed and generally don't need adjusting. Sometimes when replacing a bottom bracket we may find the fixed cup side was never faced...it happens...then we would face it because we had the tool, likely hadn't used it for some time...woohoo we get to use a cool tool...and/or we had a new mechanic who never used one before...learning opportunity. We did the same with a head set replacement...this was back in the day when the 1" threaded head set ruled the world...none of this 1 1/8 new fangled technology...if a 1" threaded head set was good enough for EddyM it was good enough for everyone...oops forgot what I was talking about...I'm old, I forget...

Titanium, Carbon fibre, etc. frames don't need facing...and honestly shouldn't even require a check "if" built correctly. I got an Aethos a couple of years ago and it was perfect out of the box. Same with the Tarmac SL7, SL8, Carbon Diverge, Roubaix and Crux I've built in the past year or so. Perfect right out of the box.

I'm still working in a bike shop, part time of course, and I've not used a facing tool in years...but we gots one.

Or after repainting, because you can't be certain that paint on the faces is an even thickness.

Oh, it's probably critically important. Of all my bikes, homebuilts, low end or high end, I've changed the cranks at least twice on every one of them, and painted painted several frames; but I've never faced a BB shell. They all seem to work OK in spite of it.

I like doing everything myself. If I don't know how, I figure it out. There is no way however can I justify buying the necessary tool that would be used once every hundred years.

Take the frame to the bike shop and let them prepare the bottom bracket shell, then take it home and assemble everything yourself.

Facing a bottom bracket falls way down the list of things to worry about.

Much ado about (practically) nothing.

If a bottom bracket threads in fine, a crank mates together easily, and the arms spin easily once everything is tightened...seems good enough for me.

The frame I built in a class was faced and chased because it needed to be after it was painted. Head tube was faced as well.
An old road frame from the 80s thst I did some brazing on had to be faced- the crank was tough to install and didn't turn easily.

I've sanded off powder coating on a couple bikes before, but every other frame I've built up thru the years, dozens plural, has been fine to just install and spin to confirm it's good.
Low standards, perhaps.

Facing the BB shell or head tube simply ensures the two faces of the shell or tube are parallel to each other while in the same plane as the threads of the BB or the tube of the head tube. It does not mean the threads are cleaned up which is why every shop that faces a BB shell will chase the threads first to ensure they are clean and the facing tool threads in without cross threading. This process does not ensure the frame is aligned.
Anyway, when I built my frames using Henry James BB shells they never really needed to be faced. After skimming off the paint layer I never had to do more than a single rotation of the tool to be shaving metal off the face all the way around the BB. His lugs and BB shells were extraordinarily high quality castings.