Can physics be done without the use of time?

Rovelli's The Order of Time is an excellent book and well worth reading if you haven't already. I don't think Rovelli is advocating doing physics without time (the article adds spice by exaggerating his position somewhat). Instead, he is suggesting that time is an emergent phenomenon rather than one of the fundamental attributes of reality. But it is a useful shorthand, and trying to do physics without making any use of that shorthand would be at best tedious and at worst incomprehensible.

In a similar way, we know that colour (in its everyday sense) is an emergent phenomenon and not an attribute of fundamental particles. However, we still talk about "redshift", for example, as a useful shorthand for "photons with longer wavelengths and lower energy", and describing a van Gogh painting without referring to its colours would be a pointless and uninformative exercise.

The question of whether time is fundamental to reality or an emergent phenomenon is not settled - which is what makes it so interesting. Julian Barbour's The End of Time presents a more extreme and definitely less mainstream view than Rovelli's. On the other hand, Lee Smolin's Time Reborn puts the opposite case and argues that time really is fundamental.