I'm going to answer your question from the point of view of interpreting what's said in the video, specifically... I think this is one of the main points of your question and I think understanding that will help you understand the wonderful world that is "who vs whom".
Let's look at what she says, first:
Many writers believe that 'whom' is on its way out because it can sound very pompous.
As you've seen in the comments, there are differing opinions on this... my personal feeling is that, in the relaxed, informal world of spoken English, this is more likely and in the more formal world of written English, it's less likely to feel pompous.
Some even believe it’s circling the drain.
Probably true to some degree. Note she says "some"... she doesn't say "it's widely believed to be circling the drain" - the idiom here meaning it's on the way out of popular use.
As I've stated in a comment, if you gave the average American a question about whether "who" or "whom" was appropriate in a series of sentences, they'd likely get it wrong. I probably get it wrong occasionally.
However, it is still used in expressions like ‘to whom it may concern’.
Yes... but it's a bit reductive to imply that this is the only place it's regularly used. I'd argue this is simply a better-known idiom, so more people are likely to know that this is an example of the correct usage.
In the sentence ‘the girl whom you’ve been dancing with is on her way to the top’
There's a couple things about this sample sentence that I feel are suboptimal:
- She's split the preposition off of the object, "whom", by rearranging the sentence.
- I, personally, wouldn't use either "who" or "whom" in this particular arrangement of the sentence.
Better versions of the sentence (in my mind) would be to either omit the pronoun entirely
the girl you’ve been dancing with is on her way to the top
or to keep the preposition before the object
the girl with whom you’ve been dancing is on her way to the top
Now, at least, including the pronoun makes some sense to me. I still prefer the version without "whom" as this version feels a bit odd or stilted to me.
I suppose the rearrangement could be an Australian construction but I'm not sure.
most writers would use ‘who’ rather than ‘whom’, and it’s fine to do that.
This is, I think, where the main crux of the question comes in. I think it's bad teaching to say "most writers do X"... I disagree that "most writers" would use the construction in her sample. Regardless of that, she says "it's fine to do that"... and that is true. It may be a technically ungrammatical choice to use "who" but it is not an uncommon actual use.
That being said, she only says "it's fine to use who in place of whom", which is generally true... what she does not say is "don't use whom".
So, in relation to the two options in your question, the answer is, "neither". You are quite welcome to use (or continue to use) "whom" appropriately. There is no reason to intentionally use "who" when you know that "whom" is correct. If you're unsure which is correct, it's acceptable in many circles to use "who" in all cases... but don't be startled if someone corrects you.
From my point of view, saying "to who were you speaking" sounds wrong... because it should be "whom". There are some great references in our network for proper usage of who and whom and encourage you to check them out for more information.