You have not 'done it all wrong' Top. You are doing your absolute best for your baby, which makes you a fantastic Mum.
Everyone, every single parent, goes through a steep learning curve with their first baby and so there will be things you didn't realise when baby was a newborn that took time for you to learn. That doesn't mean you are doing anything wrong, just that you are learning alongside your baby.
I do not think any first time Mum with her newborn wants to use a Dummy. We all expect to be Perfect Mummy, with Perfect Baby who settles without any props or fuss, sleeps wonderfully, breastfeeds beautifully, smiles all the time and cries very little.
It takes a while to realise that Perfect Mummy doesn't exist and neither does Perfect Baby.
I doubt you would find a single FTM who thinks Yes, I will force the use of a dummy and will keep going with this even though baby gags, spits it out and clearly doesn't like it. So definitely don't beat yourself up about this.
From my own experience, my DC1 was 20 weeks before I cracked and was willing to try anything to help her settle - she started solids, formula and using a dummy all at 20 weeks old (incidentally, none of which were the magic wand I was expecting), the bad habits I created were too ingrained.
By the time I had DC2 and DC3 I was absolutely certain these babies were going to sleep so I actually took dummies into hospital with me when giving birth. DC4 was the only of my children who didn't take to the dummy. I half considered not bothering when she was still having none of it at 6 weeks old, but kept going. I am glad I did.