There is a dissonance in the original sentence, even though we understand what the writer meant.
These three parts a fine by themselves, but they don't work very well together.
"is working as expected"
"by the time"
"you received this email"
We can fix this in multiple ways, such as both of your suggestions, and Bill Franke's.
Here are a few fixes with a commentary on each:
We hope that everything is working as expected by the time you receive this email.
When does the reader actually receive the email?
- When it is first read by the user?
- When it is time-stamped by the e-mail account?
'Receive' could be either one, but 'received' seems to imply the second.
So, can we keep "received" in our sentence?
We hope that everything has been working as expected by the time you received this email.
Unfortunately, the prepositional phrase "by the time" doesn't really fit very well with "has been working". So, let's try to change the preposition.
We hope that everything has been working as expected since the time you received this email.
This sounds great, but doesn't mean exactly the same thing as the original. "Since the time" implies that it started working at the time of receiving the email, but "by the time" means that it started working before the time of receiving the email.
So, we fixed the grammar, but lost some of the original meaning.
I'm not sure if we can resurrect that lost meaning without making the sentence less understandable, but here are a couple of ideas.
Beginning at a time before you received this email, we hope that everything has been working as expected.
By the time you received this email, it is to be hoped that everything has been working as expected.
In conclusion, the original sentence was easy to understand, and had a reasonably clear meaning, although it had some conflicting grammar. On the other hand, these alternatives, while they have correct grammar either change the meaning of the sentence or increase the complexity of the sentence.