Its the same thing in all CPR manuals. First check for pulse/breathing before doing anything else. Actually you should start by trying to wake them by yelling or shaking them around a bit.
This is very common in lots of emergency procedure checklists. First check if you actually need to do this procedure/use the tool/etc.
And you think you giving them CPR would not do even more damage (good luck holding them still while pumping their hearth through their ribs)? Obviously if you see them take some big hit treat them with that possibility but if you see some random person passed on the street making sure their brain is receiving oxygen is always the first priority. (after making sure the patient is in a safe place so remove them from a burning building first etc.)
Agreed, priority is to ensure there’s oxygen to the brain- but in the process of ascertaining if they’re indeed unconscious, don’t start by grabbing their shoulders and shaking them violently. Touching their hands and face as pointed out in a sibling comment is better, and after determining that they’re not breathing and have no pulse definitely proceed with vigorous CPR because potentially paralyzed is of course better than dead. Also, be aware of whether Good Samaritan laws protect you from liability before deciding to proceed.
This is very common in lots of emergency procedure checklists. First check if you actually need to do this procedure/use the tool/etc.