"She felt lonely" is bland compared to a description of how she stood apart and longed to be part of the conversation.
You've only paraphrased rather than presented the alternative that avoids the word lonely, but I agree it would be a better approach. It is not, however, the best one. Note that in "She was lonely" the last word is actually an adjective, but there's a reason it still doesn't work. (And it's not just down to whether adjectives are also problematic, which some say it is and some say it isn't.) There's a show-don't-tell issue to consider, whether you tell me she felt lonely, or tell me she longed to be in the conversation. Ideally you'd do something more like, "Alone at her table, Ellen glanced to another half-full nearby, where laughter interrupted the animated chit-chat. They wouldn't admit to having room for one more." It gets in her head more.
"She stood alone by the bar." Standing there by herself is the very point of the sentence. "She was alone as she stood by the bar" is wordier and clumsier and adds nothing to the original. Almost any other way of wording it suffers the same problem.
As noted above, changing stood to was makes alone an adjective. Is the sentence really any better? To be honest, the real issue is -ly adverbs. This book statistically analysed literature and found adverbs overall aren't any rarer in the most respected writing, but -ly adverbs are.
Having said that, it's always worth doing writing exercises that help you think of more options. If we literally ban adverbs and adjectives from the sentence, what can we do? Example: "She stood at the bar with no-one to threaten her elbow room." Whether it's better this way is a tough call!
"He leaned in conspiratorially."
Imagine that happened in TV or film. How do you know the lean is conspiratorial? What you know is how the actor moved, how their face contorted, and what they said and how. So mirror that; imagine an actor playing out your scene, then write what you see. "He didn't lean closer until looking left and right, and it wasn't for cars." The eagle-eyed would argue closer is an adverb there, but again, it's not the bad kind. (I say "argue" rather than "know" because some grammarians see adverbs as an "other" dumping ground in grammar. If you've ever caught someone saying, "actually, that's a determiner", you'll have a feel for where this goes.)
"She gradually regained her composure." Dropping the adverb changes the meaning and I don't see a way to clean that up.
She took a while to regain her composure. Well, I say a while; feel free to say "too long" or "a little too long". (Is "a little" an adverb? I guess!)
I feel that maybe I should approach it like this: Kill off adverbs for which a natural or more interesting replacement exists. Accept what you can't change for the rest. Is that a good way to approach it, or am I missing a better way?
That's exactly the way to do it. The anti-adverb advice, like much writing advice, should not be interpreted as "This rule has been announced", but as "People make certain mistakes far more often than others". If you've thought about why an adverb is still in your writing, you were probably right to keep it in. I bet at least some of the substitutions I've described herein seemed to you like trying too hard.