"Glad to hear" in non-speaking form

Solution 1:

"hear" can mean much more than perceiving by the ear. Definitions include:

• to perceive (sound) by the ear - "Can you hear that music?"
• to receive news or information (learn) - "I heard about your accident"
• to receive communication (letter, telephone call) - "I haven't heard from her lately"
• to be told by others - "I heard she got married"
• to be reprimanded - "If you don't do your homework you're going to hear from me.
• to listen with favour, assent, etc. - "She wouldn't hear of it".
• etc.

MW and TFD

In 1953, a General Service List of English Words (Oxford University Press) listed "perceiving by the ear" as accounting for 54% of all meanings of "hear". With tremendous technological advance since then (smartphones, text messages, e-mails), I won't be surprised if I hear this number has dropped significantly.

To answer your question, "glad to hear the news", "glad to hear from you" are grammatical and current usage.

Solution 2:

While you are correct in saying that hearing relates to face-to-face communication, it is often used idiomatically in written communication.

The phrase is also frequently used in writing to indicate that news has been heard verbally from a third person, not necessarily the person to whom you are writing.

In short, the phrase "glad to hear such news" is not out of place in a letter.