I am a year 6 teacher with over 20 years of experience and I have to say this is my worst year EVER for behaviour. I think lockdown affected this age group especially badly. They've been left to their xboxes for months and they have lost all manners, resilience, concentration and ability to wait for attention. Like you, I have a lot of low level disruption and shouting out. With a LOT of work we are just starting to see improvement.
First, identify your main protagonists. There will be some. Isolate them at the back and corners surrounded by your "good" ones. Ideally you want the worst behaved by themself at the back. The idea is to deprive them of the oxygen of attention. You want to minimise eye contact between the silly ones.
At the beginning of every lesson make what you expect very clear both in terms of behaviour and the standard of work. Over state it, really."During the input I expect all equipment to be out of hands, everyone facing forward, no talking, hands up if you need to ask something WHEN I say it is time for questions." Etc. If children do not do this you need to use the behaviour system immediately and say why - " I am moving you down the chart because..."
At the start of the next half term talk to them about why we have the rules. Encourage the good children to say what it feels like when they have an answer and someone shouts out etc. Then you can use this to "turn the tide" of the class against persistant offenders. "I am sorry that X ruined your chance at an answer Y, let's ignore him. You tell me your version". Never ever accept a shouted out answer, always pull them up on it and point out what an arse they have been to someone else.
Praise, praise praise the ones doing it right.
If you can, watch them at break and lunch. There will be ringleaders they are trying to impress. Figure out why.
I have shortened tasks and lessons untiltheir concentration has built up. Maths is 10 minutes of arithmetic, a starter task, input, task, input, task and so on. They can't work silently for very long.
I offer a teacher prize at the end of the week and a student choice. The student choice has to befor an act of kindness, helpfulness or good behaviour. This helps those wavering between following the idiots or the good kids make better choices. Identify those kids and specifically praise good choices. "Well done X, you ignored Y trying to distract you and your work is better as a result!"
Keep activitiessimple, achievable and controlled. You cannot control a silly class when cutting out, painting etc. You can say "When our listening is better, we might be able to...."
Once you have them 80% of the way there you can pick a more fun actvity. Immediately exclude those who cannot behave. By this time the tide should have turned and the rest should be with you.
During all of this the work for them should be at the right level. Initially, setting work was a trial. They have dropped back so much that some silliness was due to them being unable to do the work easily enough. I've had to drop some of mine back to year 3 and 4 work.
Ring home. For both negative and positive behaviour. I've also sent home "before and after" pieces of work for them to show parents.
Noise monitors can help (classroom screen, bouncing bubbles) but only if you don't have knobhead boys. For silent writing I turn lights off and put on a sound of their choice. My current class like the fake fire on youtube, my previous liked the sounds of an airport!
Use your body. Stand by those that are twats.
One last thing - because mine are so "hyper" I try to allow them to have break and lunch plays rather than keep them in. But I will keep in those who produce poor work or break serious rules. And do this consistently.
There's probably more. It's bloody hard work. I've had a decade in both primary and secondary. Year 6 are basically year 9 only a bit smaller. Good luck. And the fact that you are thinking about it is great. Also don't think that other teachers are better - they all have the same problem!