Stage 1
Find the labs worth writing to.
Nearly every research placement starts with an email to someone who has never heard of you. That is not a failure of your network — it is simply how labs are joined, at every level, including by professors. So the question is not whether to cold-email. It is which ten people to write to.
Start from a paper, not from a name. Find something you have actually read and found interesting, then look at who wrote it. Follow its reference list. Look at who cites it. Your own department’s website is a good place to begin, because being in the same city is worth more to a lab than you would guess.