Like all rules, you can break adverb if you can afford it in your strangeness budget. If you actually have rhythm (which depends highly on context) you can probably afford it. See also the rule that "editing any sentence, in any direction, makes it better".

Most adult writers seem to err by making their sentences too long. Shorter is almost always better; you just need to let the length vary except in passages with deliberate repetition.

The usual advice I have heard is that you should probably think of a better verb/adjective in the first place. For the example you quoted, "She beams" and "Her face lights up [like something]" immediately jump out at me.

Now, looking at all the look-like-adverbs in the article:

exuberantly, conditionally - I can't think of a way to merge either of these into the verb, plus they have pair structure. If not deliberately making a point I would probably change these into adjectives describing the noun'ed verb.

recently - sentence level construct, generally considered fine if you're not overdoing it (I in fact didn't even notice this until I started grepping)

foolishly, incorrectly - these modify a previously-unmodified verb that is deliberately repeated 3 times.

lovely - not an adverb despite looking like one; the other common meaning of "-ly" but unusual for taking an abstract noun

literally - generally can't merge into verbs; actually used correctly for once

swiftly, vigorously - the former has numerous words into which it could merge with the verb; the latter less. But this is clearly an example of deliberately do it for effect

swimmingly - generally can't merge into verbs, especially since it's used for effect

usually - generally can't merge into verbs, so we're stuck with it unless you rewrite to use something like "wont", "custom", ...

inelegantly, wrongheadedly - these could merge (in particular "abuse and misuse" are commonly paired), but are used for effect due to the article topic

unthinkingly - this is emphasized; "without thought" would also work. I can't think of a merge in any case, though there are numerous synonyms

pointlessly - limited merge opportunities in general, usually veering into metaphor territory e.g. "flailed". Also, this particular sentence feels like it is the whole reason adverbs exist.

early - not an adverb despite looking like one. Related to "ere" but that's the wrong part of speech?

quickly, happily, sadly, loudly - these are discussed, not really a part of the article itself

diligently - hard to merge in general

unfortunately - sentence-level

angrily - many merges exist - "shouted", "roared", "grumbled", etc. and this is one of the uncommon cases where killing "said" actually can improve the sentence. The cited "improvement" is ... actually pretty bad though.

frequently - hard to merge; has a synonym "often" which lacks the "-ly" in case you need to fool a blind rule-enforcer

silly - not an adverb and doesn't really look like one despite ending with "ly". It's actually the obsolete "seel" (good, happiness, fortune) + "-y" (resembling)

flatly - a few merge targets exist ("recited", or with some rephrasing you might use "rote") but this isn't an important adverb to eliminate

typically - hard to merge, and possible replacements might be even worse weasel words

lovingly - used as an explicit contrast structure, and few direct merges are in general, but there are many evocative other ways to express it. The article is missing a comma before it.

happily, sadly, quietly, loudly - again, these are discussed in the article itself

really, badly - this is borderline inner dialogue so the informality and simple word choice is beneficial. Many merges exist (note that since these are both adverbs you'll likely still end up with one) if you're in a context that wants them however.

loudly, rudely - discussed for style

surely - the particular shade here is of opinion, which prevents what merges might otherwise be possible

reflexively - probably can't merge, but in this sentence I definitely feel the strangeness budget straining. If this were anything but an article about adverbs I'd take a knife to it.

only - this is an adverb but not for the usual reason. Originally "one" + "-ly" by the usual noun-to-adjective construction, but has fossilized into its own idea (gaining an adverb sense) and should not be avoided. This sentence is a fragment, and the paragraph is full of same-length sentences so I'd be proper and use a comma; if I want the effect that badly then change the rest of the paragraph somehow.

weekly - this is the other other "-ly" rule, used only for time nouns

hilariously - the context is minimal but it's clear this needs to stay; using a mere pair of adjectives doesn't connect the words strongly enough. In other contexts many rephrasings are possible.

Brevity is the soul of wit.

Mostly just that it's redundant information. But there's always exceptions to any rule.

"Grins happily", feels awkward, stiff, bloated, and prompts me to expect bad dialogue. The contention is one of style. I can think of a dozen or so logical reasons why this sounds off, but for me it simply smells like bad writing.

She grinned happily.

She happily grinned.

Same words. I generally prefer the later, but they have very slight differences in emphasis. Without the adverb, the subtle difference is lost.

You could grin menacingly. Or nervously. Or reluctantly.

I know several people that grin menacingly.

You can't write clearly, without an adverb.

My favorite, plucked from Terry Pratchett:

"I'm a homosexual necrophiliac," said Tom in dead earnest.

Can you give an example? I've never noticed that (except for certain specific dialects and slang) but I may be blind to it.

I do this. I think this applies only to adverbs modifying verbs. Adverbs modifying adjectives or participles stay put.

"She runs quick," is a thing I'd say.

"***The topic was hot debated," would be ungrammatical to my ears.

Not sure how widespread it is. I think it just falls out of a natural tendency to elide utterances which don't alter the meaning of a sentence. In many positions it's obvious that an adjective is meant to modify the verb rather than a noun.

It's not a hard and fast rule. In formal writing I'd use adjectives per standard grammar.

Maybe also related to the (standard) use of adjectives as describing the state into which something is transformed by a verb. In "I painted the wall red," "red" is properly an adjective and modifies the transformative act, not the object. I suspect this construction has been unconsciously widened to apply to nontransformative verbs also.

Notably "***she quick runs" sounds highly ungrammatical to my ears.

Flashbacks to my driver's ed teacher in highschool who would say adverbs correctly then correct himself by saying it again without the -ly. This drove me insane. And no I haven't noticed this generally in US english speakers. I would assume some negative things about people who drop the -ly from adverbs.

But that's non-adverbially.

Idk, I can get behind some of it as a mechanical exercise. When I was a young'n and honestly thought I was gonna be an english lit major one of my teachers made me go a whole year without using any prepositional phrases. Did that instantly improve all of my writing? No, in fact everything I turned out under that rule was clunky and overedited. But after that year any time I ran into a prepositional phrase that I didn't like I had the experience to know how to rewrite it effectively. The point is training, not product refinement. Find a device you lean on too much, eliminate it entirely for a while, and emerge from the other side of this process with a better understanding of the crutch, knowledge of several ways to work around it and wisdom to know when to do that. It's not a substitute for taste, it's a tool for developing taste and the ability to refine things to your taste.