It does, but not when you register to vote. You are registered to a party when you go to vote in the primary (or go to a local office an swear an oath to support the party but who does that).
So, yes, it does register participating voters.
And every election year, more people register to vote in the Democratic primary than in the Republican primary.
But the number of unaffiliated voters who are registered to vote by never vote exceeds what either candidate gets. There are 18 million registered voters in Texas as of 2024. Trump got 6.3 million. Harris got 4.8.
Do the easy math on that to find out that most people in Texas don't vote.
So, no, independents don't swing the vote here. I talk to people on a regular basis who tell me they're liberal but they never vote because "Texas is a Red state so there's no point."
With this mid-decade BS redistricting that Trump ordered his cronies to do in Texas, it looks like that's going to bite him in the ass due to voter anger and that the districts are so thin no there's no real margin of safety.