can't speak to the investment accounts, but last year while the gf visited here from Philippines (ie, not a Canadian citizen), TD had no problem opening a savings account and credit card in her name with (obviously) no Canadian SIN. Her account has since been generating (very little, but not zero) interest.
I think you pretty much have to accept that your SIN (among much, much, much more of your personal information) will be "out there" if you expect to avail of their services. I wonder what the fixation is on the SIN, specifically, over everything else? You can either make peace with the fact and the use the service, or just avoid, that's basically the choice.