Highly anxious, controlling of his environment and people around him, rigid thinking etc are all asd traits. I honestly would speak to the senco at school directly rather than his teacher. My sons teacher persistently said he was "fine" the senco met him once and agreed with me, he's now diagnosed.
Even if they don't agree I would look at how to parent ND children and use some of that info to help you. For my son, punishment really does not work, especially illogical "you swore, now you can't have your bike" - it makes no sense to him, its just me being mean. We discuss what good behaviour looks like, we set rules, we reward for sticking to the rules and we apply logical consequence when he doesn't or can't (e.g. he's removed from the situation normally). I'd sit down and agree some family rules you both agree with, if he's rigid and an only child I suspect he finds it tricky to have different rules to the adults around him, so maybe you both agree it's a non swearing household, or you both agree no raised voices/shouting. There will be some things "him" specific, but discuss what is and isn't ok at his age. Write all your rules down with him, on a big sheet and stick them up in the house, agree consequences and agree rewards (think small and build up), we do a sticker or marble in a jar, ten is a Mars bar, 25 is a day trip/activity like the cinema. Make it visible.
Definitely monitor his screen time, again discuss this with him, explain the impact and appropriateness of these things. We have nothing other than a switch and it's 1 hour at the weekend. If he's rigid he needs to totally understand the rules, it needs to be totally consistent, and it needs to be written down and stuck to by both parents ideally. You cannot cave or move this boundary as it'll make him confused about what is and isn't required.
Develop a safe space for his "tantrums" personally I'd switch thinking on this and call it overwhelm. It'll help you empathise more. My son has his room, we got him an angry cushion he can throw and hit in his room that is specific for him. We agreed he can go to his room and we will let him unwind there, he has a weighted blanket he likes too. Speak to your son about what triggers the angry feelings, what helps him regain control (try breathing techniques on you tube together, candle breathing or finger breathing maybe). He needs tools to help him regulate his emotions and he needs to know what's starting the feelings (you could keep a little diary when it happens, what happened directly before, was the environment noisy, was the routine different, was it a transition period, who was around etc). Once you have triggers you can also build in mitigation to those triggers - plan the difficult situations in advance, provide quieter spaces, allow time for transitions etc etc. I think you're doing the right things, you just need to discuss and formalise them more and then stick to them and do them with him so he doesnt feel alone. Rather than a casual chat about feelings, buy the Worry monster book or similar and have proper discussions every day about his feelings, can he draw or colour his feelings today (every morning or evening). Discuss how when he is a bit older he needs to be able to understand how he feels and then work out how to handle them. I always say "feelings are the sea we are on but we sail our own boats". He needs to see that feelings roll in, and roll out, we accept what they are and we manage ourselves accordingly- you can roll model it, next time you're a bit cross "sorry ds, this flatpack has left me feeling really annoyed and frustrated because I can't do it and I feel daft as it should be easy, I'm going to make a cup of tea and sit down and have a minute of quiet to calm myself down before I try fixing it again" etc. Build it into both your lives.
It's really tricky, my son was hardest at 7ish, he's 10 now, rarely does he get overwhelmed and when he does he just takes himself to his bedroom before it gets to the stage of shouting/full blown anger. We do however, ensure his environment works for him, work out what he will find a challenge and discuss it beforehand and maintain consistency to help him. It works, you just have to stick at it.