Your first question is: What is the problem with adverbs? There are in fact three main problems.
The first problem is that they are often unnecessary. This is what Zinsser in On Writing Well: The Classic Guide to Writing Nonfiction (p68) writes:
Most adverbs are unnecessary. You will clutter your sentence and annoy
your reader if you choose a verb that has a specific meaning and then
add an adverb that carries the same meaning. Don't tell us that the
radio blared loudly; "blare" connotes loudness. Don't write that
someone clenched his teeth tightly; there's no other way to clench
teeth. Again and again in careless writing, strong verbs are weakened
by redundant adverbs.
The second problem is that they can make you appear vacillating. This is what Pinker in The Sense of Style (p43) says about "compulsive hedging":
Many writers cushion their prose with wads of fluff that imply that
they are not willing to take a stand behind what they are saying,
including almost, apparently, comparatively, fairly, in part, nearly,
partially, predominantly, presumably, rather, relatively, seemingly,
so to speak, somewhat, sort of, to a certain degree, to some extent,
and the ubiquitous I would argue.
(Pinker could have added arguably to this list of weasel words.)
The third problem is that intensifying adverbs such as very or incredibly can in fact dilute the strength of the verb or adjective they modify. Here is Pinker again (p45):
Paradoxically, intensifiers like very, highly, and extremely also work
like hedges. They not only fuzz up a writer's prose but can undermine
his intent. If I'm wondering who pilfered the petty cash, it's more
reassuring to hear Not Jones; he's an honest man than Not Jones; he's
a very honest man. The reason is that unmodified adjectives and nouns
tend to be interpreted categorically: "honest" means "completely
honest", or at least "completely honest in the way that matters here""
(just as Jack drank the bottle of beer implies that he chugged down
all of it, not just a sip or two). As soon as you add an intensifier,
you're turning an all-or-none dichotomy into a graduated scale.
Your second question is: What would be an example of a verb "with force"?
You could write: Jones produced arguments and evidence to prove that Smith's claim was false. Or you could use the strong verb refute: Jones refuted Smith's claim. The verb refute implies 'producing arguments and evidence to prove that a claim is false'.
Your third request is for an example where an adverb can be replaced with a verb "with force."
Instead of "She quickly looked over the document for errors", you could write: "She skimmed the document for errors", since skimming implies quick reading.
The term writeprint is a specific term used in the field of stylometry research. According to Wikipedia:
Writeprint is a term proposed by some forensic linguistics researchers to denote a set of distinguishing stylometric characteristics of a written text (writer invariants) such as "vocabulary richness, length of sentence, use of function words, layout of paragraphs, and key words" which allow one to identify its author (if written by a single person).
Best Answer
Some cooks hate garlic. That doesn't mean garlic is bad. That in order doesn't mean eating raw garlic before a job interview is a good idea.
The rule of thumb is "do not make up adverbs".
Do use adverbs if they concisely convey a specific quality, better than a more expressive verb would. I asseverate most people prefer an adverb - simple verb pair than an overly fancy and obscure verb containing given quality.
Avoid them when they cheaply replace a descriptive expression or a verb that pinpoints given quality.
Do not use them at all if you avoidantly try to replace whole sentences with them, or redundantly repeat the sense of the verb they describe. You are able to make up adverbs impunitively, or abuse obscurely occurring ones, or arbitrarily give them new meanings, but don't do it. It's bad.