I once thought that myself, but then it occurred to me to look at history. I could not find a single instance in the US or in several other countries that I considered to have a comparable electoral system of this actually happening. No third party has ever grown to become a real contender as far as I could find, at least not without a considerably different electoral system enabling it.

Given that the chances are close enough to zero for me to consider them such, it's clear that wasn't a viable route. Instead, every time major changes to policy have happened it's because of sufficient internal change within an existing major party, or in the couple cases where a major party has actually changed, it has been through internal division and collapse.

History thus tells us that if we want to enact that sort of change, we should get into whichever party is already closest to our position, and push hard from within, and bring as many like minded people with us to do so. It will be a difficult fight against entrenched interests that oppose such change, but far more successes have been had that way than through competing minor parties.