From jisho, it seems like 何より means best, more important than anything, above all else, etc. The meaning doesn't change much from that. I have heard ご無事で何よりです. If I use the meaning in jisho, it will sounds like "most importantly, you are save/ healthy". From the conversation though, this translation doesn't seem too cohesive so to say. In other words, it doesn't fit very well. I judge from my own guts that the more accurate translation of that phrase is really "it is good to hear that you are save (more then anything)". I realised that there it still means "more than anything" but there is also an added nuance that is (probably) more important and that is "it is good to hear". Am I right on this or is what I am saying utter gibberish that is meant for the garbage
As you have correctly suspected, this type of 何よりだ is a common and polite set phrase used simply like "(I'm) very glad to hear/see (that)". You have to remember this usage as a fixed pattern. It can still mean "more important than anything", depending on the context. Compare the following two sentences:
For children, the health of the parents is more important than anything.
I'm glad to hear your mother is healthy.
Glad to hear that.
I would greatly appreciate if you understand this.
何より ＝ Above all else, or sure most importantly
より means more than, though it crops up in places where the translation to English seems odd to our ears.
This in one good example of where simply knowing the language from a text may not assist. Why? Because if you really want to speak a language (in this case Japanese) then you must approach the concepts that you are trying to convey in the manner that the native speakers would.
So try to listen and use the phrases you hear others using, making sure you correctly understand their meaning. 何より、諦めずに、勉強を頑張ってください！