Well Heidegger would say the primary way we deal with our environment would be the ready to hand, and he gives his famous example of hammering to show his point. We don't have to analyze too closely, we just cope. We hammer without overthinking it. We see the same thing with driving and many other things we do. However, I should mention that this mode of coping "on automatic" can be potentially dangerous. We can imagine the Buddhists cautioning us to be mindful!
What Heidegger was keen to do was to draw a distinction between ready to hand and present to hand, and the
latter would harken back primarily to Aristotle and what followed from him. Typical Western categorization, analysis and so on (perhaps "analysis paralysis"), representational/ correspondence theories of truth would come in here too. Though Heidegger does not deny the importance of the present to hand, nevertheless I think he wants to show that our primary way of coping (or being) would be the ready to hand.
So Heidegger is trying to balance the Western tradition. In my opinion, and I think this is a fairly common opinion, Heidegger was a Romantic character (I refer here to the romantic era). He was suspicious of modern technology. The people of the countryside knew how to cope too without all the analysis and categorization demanded by our new scientific technological world which philosophers like Aristotle etc. had made possible. You know Heidegger had his hut up in the mountains and so on. He wrote more about his concerns with technology in his later works. I am familiar with Brandom from the internet (YouTube interviews etc) and I know he is well regarded, but since I have not read any of his his work, I can't comment on him.
As far as Heidegger ever thinking in terms of any sort of praxis or pragmatism. He did not have a high opinion of America/ Americans in general, and on that basis alone I doubt he would use the word "pragmatism". To the extent that "praxis" may be associated with the Marxists, he might have avoided that word too. I can't read German, so I don't know. I did read somewhere he was open to some sort of socialism, but I don't have a source to be able to judge how serious he might have been, or what his mature position was.
De facto, however, it could be argued that the ready to hand is a kind of practice. And the whole idea of coping with our world would be amenable to the pragmatists. But I should hasten to add that certainly the later Heidegger would be highly opposed to the instrumentalism associated with some forms of pragmatism.
In the end, he was concerned with the dangers of technology for its own sake, technology "let loose", and the contribution that typical Western philosophy had made toward this phenomenon.
