So I was struggling with this recently and... decided to go low-tech. I exported my large table with good aliases and terrible field names to CSV, opened it in notepad (yeah, it took awhile) and then used the "replace" function (CTRL+H) to get rid of the offensive field titles. Worked like a charm!
If you use the add join tool and go into Environment Settings - Fields and uncheck Maintain Fully Qualified Field Names do you then get what you want?
ArcGIS Pro has a nice tool called "Table to Excel" which allows you to export to an .XLS (if your table has less than 65535 records) or .XLSX, and it allows you to click a checkbox that states "Use field alias as column header". However, that checkbox is referring to the field alias registered with the feature class, not the table inside the map, as you see it. So, if you can use the field aliases as they are set in the feature class, that's a great tool to use. But if you've modified the field aliases in the table in your map, and then want those new ones, this tool won't work.
However, if you're dataset isn't too large, you can still select all records, copy them, and then paste them directly into Excel. That preserves the alias column names you customized, as you see them in the map table.
TL;DR: You can use Pro's "Table to Excel" tool, -OR- you can just select all records, and copy/paste directly into Excel from Pro.