How to discipline?

Our Sun conure has a biting problem​​​. Every bird bites, it's natural and the past two years we had him he did too. However in the past when he would bite it wouldn't be very hard, just a quick squeeze when he wants/doesnt want something. Sometimes the bite would be hard but it was only ever when he would be trying to climb or pull himself up.

Recently however I believe he's figured out that the harder he bites the more results he gets, and that's what hes doing​. In the past if he would start chewing or playing with something he's not supposed to, lets say a wire, and we would take it away he would at most let out a little yell and then move on to a toy. Now however he'll scream, twitch his wings, try and look big and try to lunge at our hands, and if he catches them he will bite very hard.​​​​

I know that to try and correct this the best thing to do is to put them on their cage or a surface and walk away/ignore​ them so they learn theres no use, but the problem ​​is that hes very clingy and has wings, so he just flies right back to us or sometimes just refuses to step down. Ive tried putting him in his cage for time out but I don't want continue because I don't want to risk him no longer liking his cage. ​​​​

Where do I go from here? Is there a different way I can​ discourage​ the hard biting? When he flies back to me should I just keep bringing him back to his perch​ until he stays? This is the first bird I've owned, so I'm really at a lose here.