No, they demand a principled balance between the two freedoms, that's independent of the content of the speech. What you're talking about is what's status quo - freedom for private businesses to not serve someone, except when it contradicts certain narratives, at which point it becomes a national news story about bigoted business owners.
Saying the balance should be predicated on ignoring the content of the speech makes sense only if speech itself was a pure function without side-effects, only if saying something truly had no other consequence than it being said can we safely ignore the content.
But the precise content of speech is exactly the part that matters! If a death sentence is read out aloud in verse, or just sent in via a short text message (that is via two very different types of speech) is clearly not very important if the end result in either case is an aid worker being blown up with a Hellfire. Speech very often triggers actions, to ignore this fact is to state nothing may be prevented because judgment is only possible by examining actions committed to reality.
The "balance" is based on something that, while explicitly stated, you somehow disregarded: that some--and, by and large, the current legal landscape--considers the freedom of speech right of the platform to censor people, and so to obtain "anyone can say anything" we have to take away "the freedom to not say something"... which we actually already do in various instances (often involving protected classes) and so, personally, I consider that to be a tractable goal to regulate: some things are "utilities" or even "quasi-governmental entities" and should be forced to be something like a common carrier.
