Yes - I can't find anything grammatically wrong, but certainly your second example doesn't sound natural. It's probably, as you imply, that a lot of information is being condensed into a small space, when it might be better put in extended constructions (get a move on with those prepositional phrases!) or even more than one sentence.
She waited outside this morning.
She waited impatiently outside.
??She waited impatiently this morning.
She waited impatiently outside the school this morning.
She waited - impatiently - outside, this morning.
She waited outside this morning. Impatiently.
What is the order of adverbs?
Because adverbs are used to modify verbs, adjectives, other adverbs,
phrases, clauses, or even entire sentences, they are able to function
nearly anywhere in the sentence, depending on their type and what it
is they are modifying.
If we use more than one adverb to describe a verb, though, there is a
general order in which the different categories of adverbs should
appear—this is known as the order of adverbs (sometimes called the
royal order of adverbs):
Manner
Place
Frequency
Time
Purpose
Of course, it is uncommon to use five adverbs in a row to modify the
same word, but if a sentence uses two or three, then it is best to
follow this order to avoid sounding unnatural....
(*Note: For the sake of conciseness, both single-word adverbs and
adverbial phrases will be referred to together as “adverbs” throughout
this section.) ...
[I]if we were to make a sentence with all five categories of adverbs
together, it might look like this:
“I have to run quickly (manner) down the street (place) each morning
(frequency) after breakfast (time) in order to catch my bus to school
(purpose).”
Even though the string of adverbs is unusually long, the sentence
still sounds smooth and logical because the order is correct. Now
let’s try rearranging the order of the adverbs:
“I have to run each morning (frequency) quickly (manner) after
breakfast (time) in order to catch my bus to school (purpose) down the
street (place).”
By changing the order of the adverbs, we’ve actually changed the
meaning of the sentence, or at least made the original meaning nearly
incomprehensible. This is especially apparent with the adverbial
phrase of purpose 'in order to catch my bus to school'— by placing it
before the adverb of place, it now sounds as though it’s the school
that’s down the street. There is not such a drastic shift in meaning
for the adverbs of frequency, manner, and time, but they still sound
awkward and unnatural in the new order.
When we can change the order
There is a great deal of flexibility regarding where in a sentence an
adverb can appear, regardless of its content and the rules of order
that we looked at above. While the order of adverbs is useful to keep
in mind, it is a guide, rather than a law.
..........
Multiple adverbs of the same category
When we use multiple adverbs of the same category to modify the same
verb, we order them based on how specific the information is that they
provide. For example:
“On my father’s ranch (place), I often (frequency) helped gather the
animals at the end of the day (specific time) when I was younger
(non-specific time).”
“I lived at home (more specific place) with my parents (less specific
place) to save money (purpose) while I was working on my doctorate
(time).”
And if the Royal Order ruling has to be seen as a rule of thumb, it is almost inconceivable that this codicil is more binding.
Best Answer
According to NOAD, the definition of home in this context is given thus:
Technically, it is not ungrammatical to say The children are playing home to mean The children are playing at home, since at is implicit in the definition of the adverb home. However, this form is certainly not colloquial, and I daresay no native speaker would use this in any context, formal or informal.
Here are some examples in which it is all right to drop at (informal contexts):
In the following examples, it would be incorrect to include a preposition:
You may notice that home is generally used alone as an adverb of place (without at, from, etc) with movement verbs, e.g. come, go, leave, stay, be, drive, dash, move, etc. Verbs of other species usually require at, e.g. dance, play, eat, relax, and whatever else one does at home!
To my ear eat home sounds all right, and I may have heard or used it myself. This may be one of a few exceptions, but it is certainly not formal. If eat home is ever used, my hunch is that eat in (which means eat at home) is falling out of favor, or, as native speakers tend to do, at is dropped for convenience and it sounds good!
Another observation to note is in the case of non-movement verbs, the preposition may be dropped (colloquially) when the action described can be performed at home as opposed to somewhere else. Examples are few and far between, though.