Making a site Visual Editor friendly

I recently had my site converted to WordPress but when switching between the visual and html modes all line breaks were getting stripped out. This was partially solved by wrapping all br tags in a class. If I re-enable wpautop I can successfully add ‘Enter’ paragraphs via the Visual Editor but ‘Shift-Enter’ line breaks do not get saved.

I’d like to do away with this workaround so that going forwards I can make the majority of straightforward content updates via the Visual Editor. Will I be wasting my time if I go through all my site’s code and replace all br and p tags with the appropriate white space or is that the best way to make my legacy code WordPress friendly?