Yes - You say that, but [some refutation or caveat] is idiomatically commonplace for many native speakers, in response to an assertion they disagree with.
It's nearly always used in a context where although the speaker doesn't fully agree with the preceding assertion, he recognizes that there's some truth in it.
In fact, you'll often hear You could say that, but..., which more explicitly acknowledges that the assertion is at least "credible", even if it's not completely accurate. Note that in this usage, you = one, anybody, not necessarily the person being addressed (i.e. - what it amounts to is something like It wouldn't be ridiculous for you/someone to say that, but it's not entirely true).
Also note that if the personal pronoun is stressed (You say that, but...), the implied meaning shifts significantly. This form almost always occurs in contexts where the speaker completely disagrees; he intends to dismiss your opinion on the grounds that you're ignorant or biased, and/or to cite other sources more authoritative than you to refute what you said.
There is nothing wrong with climb down, especially since it's commonly used.
As you have found, it is common enough that it has dictionary entries. Here is another for climb:
climb
3. : to go about or down usually by grasping or holding with the hands <climb down the ladder>
Presently, this is how the verb climb can also be used. Contrary to what your source says, there is nothing imprecise or ambiguous about climb down. Choosing to use it or not use it is a matter of taste, as he admits.
Just like go up and go down the ladder, climb up the ladder can also be used. This is not incorrect and not necessarily redundant.
I glanced through the thesaurus and the only close alternative I found was descend. If you tell a kid "Descend from the tree!", it would sound strange and out-dated (if people ever really spoke like that).
I think come down is a good alternative, as you have suggested. You can also use get down, as in "Get down from there!"
Best Answer
Is it gramatically correct? I'm not sure. A computer is not something that can be played. You can play on the computer, and you can play a computer game.
Is it acceptable to say? Absolutely! I say "playing computer" all the time. Actually, I probably say this more often than "playing computer games". Everybody will understand your meaning, and I doubt that anybody would think it sounds strange.
Now, StoneyB has a good point, if you say "playing the computer" (or even "a computer") it implies you are playing against a computer. This would sound a little bit strange unless someone is asking who you are playing against.
These all sound perfectly fine.
Both of these also sound fine.