I would like to brine a brisket for pastrami but would prefer not to use curing salt. Does himalayan salt in brine make brisket pink as if using curing/pickling salt?
No, not really. There are plenty of curing recipes for pastrami that do not use curing salts and you can use Himalayan for them. It is really just rock salt with some iron oxide (rust) in it. Some people claim an improved taste, others say it is imaginary, but it is a substitute for table salt, not curing salt.
As said, plenty of recipes do not use curing salt, and you can make a pastrami you may really like. But the more vivid pink color will not be there. That color comes specifically from the reaction with the curing salt nitrates and/or nitrites, not the pink coloring added to many of the salts. That coloring is added as a warning that this is not table salt, don't confuse it and use it as table salt. The current popularity of Himalayan is confusing some people and could cause accidents that need to be avoided.