You can definitely do that, and it will work. A unique index will still force sequential operations, but only on that SQL call, rather than the entire usecase, so the likely hood of failure is much smaller.
The only issue is that you have to handle failing usecases. Say the email was written to the read model at the start of the usecase, but some business operation failed and the event was never actually stored in the log. Now you have a read model with invalid data.
If your event log and constraint projections are stored in the same SQL DB, you can solve this with a transaction around the entire usecase (we do this).
There is still the issue of potential failure. Ie. too many requests hitting the DB, but it's not one you need to deal with at the start. You'll only face this when you have massive numbers of people registering at once. We're currently adding monitoring for this, just to warn us if we're reaching that threshhold. It's unlikely we'll reach it, we're not building Facebook/Uber, but it's reassuring to know it's there.
Yes, I was assuming same database as it is the only convenient way to get full consistency. With separate databases, you pretty much have to use eventual consistency. (Well, distributed transactions exist too, but become problematic under load.)