Well: up to a point. Here's something I'd love to see on this forum (I'm not holding my breath) - a season that passes without people posting that School A has won because of its amazing spirit, whereas School B lost because of its bad attitude.
As a general rule, School A wins because it has better players, or is better prepared, or (very occasionally) has a lucky bounce of the ball (actually, if we're honest, the ref sometimes comes into it, too, but let's not go there). My observation of many decades of school Rugby is that pretty much every team that goes out there on a Saturday afternoon is committed and makes a genuine effort.
Team sprit is a strange thing. It can, sure, be developed in teams that pass through age groups together for six years. But it can also develop over intense training sessions and matches in the course of a single season. When I was in Year 12 at school, we had a few players who turned up at the school for the first time that year, who made the 1st XV (not, I hasten to add, on scholarships - back then it was quite common for parents to send boys to private school only for the last two years, as it was thought that this would give them a better chance of a decent HSC). I don't recall them being treated differently to anyone else, or contributing differently. Look - when you're on the bottom of a ruck, in the same coloured jersey, you don't think too much about the path you took to get there over the last five years. You're in it together.
Does that mean that team spirit counts for nothing? Not at all. A determined, spirited team will (very occasionally) upset a team with superior talents and athleticism. It's part of what can make the game great to watch. But, if I'm honest, if I were a coach and you offered me the choice between a so-so player who had been part of the group all the way through school, and the guy who played the same position for Australian Schools last year, well.