Does the notary provide the title insurance (as in: if it turns out later that the seller didn't actually own the house, the real owner shows up and proves it, the bank and new buyers have their money covered)?
That would make more sense for the 2% but it's not related to the notary in the US; notary literally just does "I checked the id of the person signing and it matches" which is a service that can be had for free at your bank or at most $50 if you have to pay someone, 2% of a house for that service obviously wouldn't make sense.
In Germany you cannot sell real estate thatbisn't yours, ownership is registered. The notary, as part of his job, makes sure this title transfer is done properly. If the seller isn't the owner, the sale is void, as far as I know sales price is to be paid once the title transfer was requested and approved, upon proof of payment the notary finalizes it. So basically no title insurance needed.
By the way, excerpts of the registry for the real estate in question are part of the sales contract (just remembered).
Ownership is registered in the US and part of the sale is a "title search" to check that and then update the registered title when the sale is executed (both handled by an attorney).
The case of title insurance is only where something weirder happens with wills, contracts, or liens. If there was an owner that died in 2016 and his will said "split my house to all surviving children". And at the time one child shows up who believes they are the sole surviving descendant and claims the house and sells it in 2017, then 2018 a second heir finds out their estranged parent died and shows up to say "actually half of this house was mine, that guy didn't legally have the right to sell it even without any fraud occuring".
Title insurance ends up 'fixing' this problem so that the buyer and actual-should-have-been-owner can both be made whole (and they go after the seller for the amount they didn't actually own, but sometimes that won't be possible to recover). If you don't have title insurance you as the buyer just take the liability that you might not actually own the house (or part of the house) if the record was mistaken and needs to be corrected.
In Germany would this just be a case where if the will is executed and you find out too late the inheritor would be screwed with no way to get what you owned instead?
A notary in the UK is a type of lawyer (although not necessarily a solicitor) and requires special certification and training. Their job is to make sure all the paper work is in order and to sign off and legally certify that is in fact so.
That would make more sense for the 2% but it's not related to the notary in the US; notary literally just does "I checked the id of the person signing and it matches" which is a service that can be had for free at your bank or at most $50 if you have to pay someone, 2% of a house for that service obviously wouldn't make sense.