I have an almond tree (same family and very similar care) which required very similar care after having being neglected while the house I now live in was left empty.
You must completely remove that branch, or the fungine infections which killed it will end up killing the whole tree. This must be done during the Winter, while the tree is dormant, or the tree will ooze resin and sap which may cause even more health problems.
Do not cut the branch flush to the trunk but leave about one inch. This wound must be sealed immediately after the cut. Personally I favor a homemade mixture (in weight): 50% vinil-based glue, 48% water and 2% copper based fungicide. Stir well, then generously apply to the cut. During Fall you will remove it to check how the cut is healing. If need be you can reapply the mixture and leave it for another year.
I totally disagree with every recommendation you have made here.
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