The main reason why a military circuit is 'racetrack' shaped rather than rectangular is that a 'racetrack' circuit is more appropriate for a low wing aeroplane - in the 2 continuous turns you have a good view of any other traffic inside your turn. Whereas a rectangular circuit allows highwing aircraft to conduct 4 quick turns, during which the wing effectively blanks the view of other traffic ahead. Whereas 2 continuous turns would mean a longer period of blindness.
Attitude flying is a core aspect of all correctly taught flying training. 'Select, Hold, Trim' is taught right from the start and is not unique to civilian training.
Civilian circuit patterns do not necessarily require the pilot to level off before starting the crosswind turn as you seem to think - the local circuit pattern may have noise abatement requirements though. These will be forced on the aerodrome operator to the detriment of proper training - whereas the military rarely accepts such nonsense at a flying training station.
Flying an accurate speed with a fixed-pitch propellor is far more difficult than with a constant speed propellor due to the effect of IAS on blade angle of attack requiring the pilot to control IAS in level flight with indirect corrective use of the throttle lever. Whereas with a constant speed prop, power can be controlled directly by adjusting MAP with the power lever and the prop corrects as required to maintain the set rpm. Hence 'set 2250 and accept the speed' is often easier and safer (more time for l00kout than chasing ever-elusive speed fluctuations every time vertical air movement is encountered. Flying an accurate speed in a jet is really much simpler - just the one thrust lever per engine and no damn propellor to screw things up!
The run-in-and-break, flown correctly, is a very safe and practical way of entering a visual circuit. The original notion meant that a military aircraft could maintain tactical speed for as long as possible before scrubbing off the speed in the break and deceleration to final. For example, allied tactical air force operations from a forward aerodrome were often threatened by lurking enemy fighters in the latter days of WW2 - so pilots needed to maintain fighting speed for as long as possible. The unfortunate design of the otherwise excellent Me262 obliged it to fly a long approach as it had no speed brakes and poor engine response - so slowing the thing down to a safe approach speed wasn't easy. The Luftwaffe lost quite a few 262s on the approach to allied fighters and resorted to covering the approach route with intense AA cover.
Other military differences? 'Point and power' visual approach technique which is way better than the technique taught to most PPL students. But if taught to PPL students, they usually solo an hour or so earlier as it is such an obvious and intuitive method of controlling the approach. Another difference is 'Standard Closing Angle' on visual navigation exercises - which makes cross-country navigation so simple that many old dinosaurs won't use it; they prefer copious unnecessary mental arithmetic, it would seem.
Military flying training teaches pilots to operate their aircraft confidently to the corners of the approved flight envelope - whereas PPL flying merely teaches students sufficient skills not to kill themselves when flying from A to B for a £100 cup of tea.