I assume that Overground gets inspections because so many stations are without barriers. When commuting on the Met line there were occasional on-train inspections before barriers were installed at the outer stations. I never saw one after that.A senior rail card discounted out boundary travel card from Martins Heron is worth it if you are doing more than one Zone 1 journey. As for on board ticket inspections I was checked yesterday on an Overground train, two burly ticket revenue chaps going through the carriage. I seemed to be the the only one with a paper ticket, the rest were a combination of oyster/contactless/phones so regrettably the writing may be on the wall for paper travelcards.
Apart from track bashing enthusiasts there are a tiny number of edge cases where a paper ticket would be advantageous for use within the zones.