Most rental spamcans are knackered, and most maintenance companies don't understand the "technical" stuff like fuel injection. Their expertise is so often limited to squirting aerosol oils into bearings.
As a long time mechanic and inspector, I've never personally seen a shop filled with mechanics ("engineers" to you) so stupid and inept that they are unable to maintain an injected engine, or who find it too "technical." Surely you must have found the most inept on earth.
I've worked for several companies instructing and flying charter and flying maintenance in rental airplanes, and haven't flown any that are "knackered," or maintained by those who can't fathom the increadibly simple fuel injection systems that the airplanes use...or carbureted systems, for that matter.
You truly know mechanics who can do nothing more than "squirt aerosol oils into bearings?" Report them immediately...and then trap them under glass. You've discovered a rare, primitive breed, the likes of which is seldom to be found. Most I know are professionals who do far more than "squirt aerosol oils into bearings."
IO. I can see an emerging theme from your recent posts and I think I can see the sense in this.
As can I; a strong theme of arrogance which presupposes a lot of incorrect information. Some might call it self-righteousness.
Pumping the throttle does two things; it opens the air valve known as the throttle, and on some engines it activates the accelerator pump. It does NOT prime the engine.
If you need prime, use prime.
If the manufacturer recommend method for starting an engine which is heat soaked isn't working, then you may either be doing it incorrectly, or you may have components in need of replacement or adjustment.
Use caution moving or being around the propeller when the engine is hot. At all times, really, but I've seen some who feel that if it won't fire off right away with the starter, then they're okay to move the propeller by hand. I've seen some who flood the engine, then pull it through by hand for some inexplicable reason, prior to hand cranking again. Don't do it, if someone recommends the practice to you.
If the engine isn't firing, consider what makes it fire, and look at what you can do to change your choices. The engine requires fuel, air, and spark. Some engines are started on one magneto because of a shower of sparks system or impulse coupling and to prevent unwanted action by the other mag due to a timing difference or for other reasons. Note the requirements of your engine installation as spelled out in the aircraft flight manual and any relevant supplements. If you have a weak spark plug, with an incorrect gap or too high resistance, or slightly or largely fouled, it may be affected by the temperature increase or the change in location (launching from a different elevation, for example). Old spark plug leads experience a change in resistance when exposed to heat, as well. A magneto, condensor, coil, and other components may also experience changes in properties in the presence of heat...these are things you can't control as a pilot, but which may prevent spark. Corrosion in spark plug leads or attaching components, burned and pitted points, etc, also affect this, as does incorrect internal mag timing or mag-to-engine timing.
Airflow can be impeded by a dirty filter, or the formation of carb ice...which can form even during the engine start under some conditions. If you're unable to obtain a start through the normal filtered air, try it with carb heat on or your alternate air. It's one change you can easily make which may on some occasions make a difference.
You can help vent some heat from the engine by leaving the oil filler door and other doors open on the nacelle or cowl when you're away from it...but use caution so as to not forget the doors and lose one in flight later. I've seen that happen a few times. I've also seen folks cover the intake as part of a post flight, only to forget it on start up. The engine needs air to breath, just like you and me.
Fuel...there exist a lot of possibilities...from contaminated fuel to plugged injectors to improper use of your fuel controls...throttle (which is really an air control, in many installations...not a fuel control), mixture, the fuel selector, primer, etc.
If the engine isn't starting, something isn't right, be it a mechanical discrepancy (as alluded to by IO540), a fuel discrepancy, an air discrepancy, or a spark discrepancy. The mechanical you can't immediately address, but you control when spark is introduced, the amount and placement of fuel, and the airflow.
Simple solutions are to go to the extreme. If you suspect not enough airflow, an open throttle will provide the maximum. This also presents some hazard. If you're operating carbureted, it also rejects the idle throttle jet...no way to get idle fuel flow when the throttle is open, and no way to get adequate main jet fuel flow without adequate flow through the carburetor venturi...so simply opening the throttle without a means of providing additional fuel from another source...the accelerator pump is an often-abused method. The primer is another source. The open throttle does, however, provide the maximum amount of cooling airflow to be drawn through the combustion chamber prior to the introduction of fuel. It's also an excellent way to get a backfire or an afterfire when starting with fuel being introduced, so use caution. A backfire can dislodge your induction or cause an induction or carburetor fire, whereas an afterfire can damage your exhaust or cause an exhaust stack fire.
Additional prime may also help the problem, but too much can contribute to a fire, if not used judiciously. The introduction of prime while cranking may be a help, as it presupposes an increased airflow with the partially open throttle. there are two ways to start with respect to airflow and fuel flow; one is on the mixture, the other is on the throttle. One means with the mixture preset, and advancing the throttle until the proper mixture is reached (such as starting on a flooded engine; the present mixture to rich provides ready fuel flow, and advancing the throttle gradually slowly admits more and more air until the engine fires). The other method is starting on the mixture, with the throttle preset, the mixture at cutoff, and slowly advancing the mixture until the engine fires. Both methods may be done with or without a flooded engine, but can be used to take advantage of a slightly flooded situation.
Applying additional fuel, or flooding prior to cranking means that fuel is present prior to the spark. This is a generally undesireable condition; spark should usually be present prior to advancing the throttle or providing something in the engine to burn.
Applying primer to a hot engine isn't necessarily a bad thing, though it's typically thought of as a cold engine tool. It's not. It's there to be used as necessary.
Think through the firing process before arbitrarily trying a random starting technique. Your engine will have prescribed procedures for starting under varying conditions, but if these aren't working, reason out why it doesn't appear to be starting, and then address each of the possibilities; fuel, air, and spark.