I have a lot of sympathy for you. Being somewhat older than you I hope you don't mind my sharing some personal experience
I too hate condoms. Can't feel anything dont trust them, very talented at squeezing them off so ive never known them to be effective.
I also had my 2 DC in my very early 20s and married. I too don't do well on hormonal contraceptive and by then I'd had one Copper coil go walkabout. I knew I was done with 2 DC. My husband went to get his tubes tied. NHS would not do it so he went private, our last DC together was less than a year old. He had to do some persuading (10 years together, married, two children, failed contraceptives etc)
Guess what? Less than 1 year after that we were very suddenly split. 2 years later hed started hard core dating aka new wife searching. Roughly 3 years later he'd had the op reversed 4-5 years later he had another child. Eventually he had another 2 and ended up dropping contact with our DC.
I never had anymore. I had another failed copper coil. I also I had a wobbly in my mid thirties but because the hormonal contraceptive I'd resorted to can take up to a year to stop working so I thankfully I thought the better of the idea before the protection actually ran out. At 40 something now I can't see my changing my mind.
Now I'm not saying you will end up like I did, but I do now understand why they say don't do anything young. You really don't know what will happen and you have too much time for it to happen in. Men (and woman) seem to solidify a new relationship with further children. There's no harm in asking him to have a vasectomy. but I do think you need to prepare yourself for him to say no and that, horribly, it's the right decision for him.
As it turns out I really could have had my tubes tied in my 20s and it been the right choice. There's not way I'd advice it to my younger self though. You'd be desperately sad if you did at some point decide you wanted further children and the operation didn't reverse properly, I was sad enough when I realised the Depo couldn't reverse immediately I I thought I wanted another one.
No I'm not sure where that leaves you. In his shoes I probably would prefer celibacy in the relationship to a permanent contraceptive solution. I dont think its a given he will cheat, tbh id worry as much about unsustainable levels of porn use as much as cheating. But you come back to the fact you really don't know what will happen tomorrow and being young, you have a lot of time for the unexpected to happen in.
Try to have a honest conversation But he probablu won't. To do so would mean him having to admit he's given serious thought to his own interests when/if the relationship ends. He's not going to want to hurt you like that.