I am still building a chute ...I consider it bad advice when I got started 4 years ago that I didn't need a chute or a squeeze. True, they aren't as necessary as a water trough or a fence but there are times when they really help, and even if its once a year, the safety it introduces is significant.
For all things chute, Temple Grandin is the person to listen to. She has a bunch of designs for large operations ... I have a herd of 8 so I was pleased when I found this presentation: https://www.hobbyfarms.com/corral-cattle-herd-build-temple-grandin/
I'm still building mine so I don't have any learned comments, although I am making a Y split in he curve with one path leading to a squeeze (and back to the holding alley) and one leading to a loading ramp.
As for a squeeze, new metal ones are really expensive and they all seem built to handle rodeo bulls all day long. I went with a simple headgate instead - still steel. My cattle are quite chill.... we'll hope that is enough.
Generally milking parlors have smooth, hard floors for cleaning and such. But if you're outside that may be less of an issue. I think you are still at greater risk of contaminating the milk, but you can perform your own risk assessment. Its easy to make a little wooden headgate - you want to encourage the cow to hang out and get treats, and you're not imprisoning her. You will need a significant excluder fence/bumper to keep the cows from cheating!