How is rollup cheaper for non-smart contracts, one shot transactions?

This has been confusing me for a while. I can understand that rollups are cheaper for smart contracts, because they don't execute contracts on chain, but only do it when challenged (optimistic) or only doing the proof (zk)

I can also understand that they're cheaper if I stay in L2 for a while, making multiple transactions, and eventually they can be batched to lower everyone's cost.

What I can't understand is how this works for single shot transactions. Say I'm transfering X amount of eth from L1 address A to L1 address B through rollup, I imagine things go like this

1. I give the rollup mainter a transaction signed by my private key (off chain)
2. Roll up maintainer gather more transactions and batch th, compute the new l2 state (off chain)
3. The rollup maintainer call a smart contracts to
   1. update the inner state's Merkel tree in blob
   2. actually execute the transaction

Now 2.2 is what confused me. I can't imagine how it actually execute my transaction, without posting my signed transaction to the chain. If it does so, why is it cheaper than me posting it on chain myself?

I know that rollups can post some commitment of my signed transaction, but how can Ethereum execute a transaction with only a commitment? My money is not even in the smart contracts in the first place.

Thanks in advance! And please kindly point out if there's anything wrong in my assumptions.

ETA

People tell me it's not cheaper, which makes sense to me. However, I got this question because lately when I want to withdraw from an exchange, it offers 'Arbitrum' which is significantly cheaper

https://imgur.com/a/yVhObF8

The receiving side also offers Arbitrum, but the fees aren't shown there (which makes sense)

I suspect it's a marketing trick to subside my fee to make me locked in? I didn't try ot out though, because I don't feel comfortable transfering money through something I don't understand.