RV7A pilot reported an NMAC with another aircraft in traffic pattern.

Narrative

Returning to ZZZ from the east; the pattern appeared empty; so I elected an overhead pattern entry to downwind for Runway XX. One aircraft was waiting for departure; Aircraft Y a Cherokee 140 who announced she was holding short for landing traffic. I never heard a call from the landing traffic. I called 10 miles east; 5 miles; 3 miles 1 mile; and overhead and intention to land entering downwind from overhead. Aircraft Y waited for a few minutes for landing traffic. I saw Aircraft Y on ADS-B; but the landing traffic had no ADS-B. Aircraft Y departed when I was at 3 miles and she called a left downwind departure to the southwest. I tracked her departure occasionally watching the ADS-B target.When I called overhead; I heard a call from Aircraft Z; a Cherokee 180. This is a flight school training aircraft with NO ADS-B out. The call was simply 'Aircraft Z downwind'. I heard no call from Aircraft Z the entire time I was approaching to enter the pattern for landing. I then saw Aircraft Z directly in front of me passing right to left at about 1/4 mile soon after I crossed 34 at midfield. A collision wasn't likely; but I made a 20 degree right turn to pass behind; then swung outside the downwind making a left 120 degree turn to fall in behind Aircraft Z about a half mile. I turned left instead of teardrop right to keep Aircraft Z in sight. I hadn't yet gotten down to pattern altitude which is 5700 ft. I estimate I was 200 feet above and about 500 feet behind as we passed; but I had him in sight the entire time and let him know. As soon as I saw Aircraft Z; I announced that I had him in sight. Shortly after; either the pilot or CFI announced very loudly 'did you see us?' or 'you almost hit us' and I replied 'I replied 'I announced that I had you in sight'; then he said 'we're Aircraft Z; not Aircraft A and very firmly 'that is not okay'; or something to that effect. I figured I startled him and thus his strong reply. I may have said Aircraft A instead of Aircraft Z by mistake because the other departing aircraft was Aircraft Y and both are owned by the flight school. In fact there's another Aircraft A owned by the same flight school that's a Skyhawk.I didn't mean to startle the pilots; but the situation could have been avoided if Aircraft Y had made sufficient position announcements; especially knowing the flight school has no ADS-B out in that airplane.It's interesting to note that only yesterday; my hangar neighbor who is a CFI and was flying Aircraft Z had a stuck mic that he eventually cleared by plugging in his own portable mic key. It is quite possible that some attempted calls by Aircraft Z could not be heard because of inadequate maintenance of Aircraft Z. Furthermore; a flight school should install ADS-B out in their training aircraft. I know it isn't required in our area; however; flight schools should invest in them for training aircraft in my opinion. The flight school has been notified about the potential defective mic key in that aircraft by my hangar neighbor.