"It shouldn't matter if a Democrat is voting Republican or the other way around. People can change their minds, and should not be disallowed to vote if they find that the other party is doing a better job in making the country a better place."
This is the Primary. Not the actual election. Those people are free to vote for whoever they want in the general election.
"If a whole bunch of Republicans vote Democrat, it means that the Republicans have failed to do their job!"
In the general election, yes. Not in the primary. In the primary, it means that one party is trying to choose the candidate in the other party that would be easier for their party to beat.
" Saying you can't vote because of that is like a childish loser saying you can't play for the opposing team."
No, it isn't. Not when you realize what election you're talking about.
"My point is that such a decision should not be made and it segregates our democracy and makes it an "us vs them" fight, when in reality we're all in this together."
Your point doesn't make sense when a party is trying to pick it's own candidate.