"She grins happily". Sure, "She grins" also conveys the same thing, but the two sentences differ in word count.
I personally feel that reading has a rhythm to it, and adding more filler words just to make it coherent with the surrounding sentences isn't bad... fictions, at least...
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.
Moving "only" to any other spot in this sentence is grammatically correct and changes the meaning.
As for "rules", well, welcome to English: the rules are made up and the points don't matter.
11 characters. Could omit the period and spaces for 8. Needed to use the meme spelling "smol" since it's shorter than "small". Can we get it shorter without losing the meaning entirely? Maybe `wit□smol` though that's not the syntax for how `□` is used in modal logic, and still 8 characters (though a bit closer to the original meaning).
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.