The question asked is,
Is it a common business practice to let one's personal assistants manage one's email?
Yes. It is both technically possible, and common, for publicly-visible individuals (CEOs, partners, etc). within a company to have help from an admin assistant or PA answering emails from the public.
The PA role is typically responsible for filtering and focusing: Helping the senior employee provide the most effective input on the most relevant topics/problems. Often, part of this role is handling "easier" tasks as a representative of the senior employee - things like opening mail, managing calendars, and responding to basic questions with pre-approved messages.
From a technical perspective, it's easy in most email systems to set up delegation: The senior employee allows the PA access to their mailbox, for the purpose of performing this role. The messages may still come from the senior employee's address, and there may be no indication that the PA sent the message. This is the modern-day equivalent of an admin assistant typing up a letter and either having the senior employee sign it, or rubber-stamping the signature.
In less-saavy environments, this may be done via sharing the senior employee's password, though this is not exactly best practice.