Unlikely. The military is set up to operate in extremely volatile environments where tiny mistakes can have immediate consequences far more deadly than anything the police ever faces. There are far bigger threats to them than someone else shooting first. Resisting immediate fears for the sake of bigger, more abstract goals/dangers has been a key element of military training ever since the ancient days of hand to hand field battles and this is just the continuation of that.
The police, on the other hand, can easily give in to the immediate fear of someone else shooting first: when they are not careful about not escalating, the worst thing that is likely to happen to their side is having to call in reinforcements and having some more bad press that won't add much to whatever they already have. If you want to put it in very crude terms, they are playing war in easy mode and there, many things that are very important to the military are completely irrelevant.
An opinion from a non American: This is what you get when you focus on tactics without focussing on strategy. Reactive vs proactive. By splitting the American police into many many different layered jurisdictions with no overall plans or even goals, there will never be a coordinated strategic view of anything.
The US military is far more centralised and (despite its many faults) aware of planning for wider strategic issues.
Many police depts don't seem to want to face up to the fact that the tactical problems they have now have root causes in their own short sighted policies, incentives and behaviour. If they carry on like they are, these problems will only get worse.
Other western countries with more centralised police hierarchies seem to put more effort into community relations and other more strategic stuff, even if it means less 'effective' tactics ie less control, arrests and convictions etc.
That does make a lot of sense. This reminded me of battle of adrianople, where the roman army was gathered for negotiation purposes and the emperor was not read to attack. But some Roman units began the battle without orders to do so, believing they would have an easy victory, and perhaps over-eager to exact revenge on the Goths after two years of unchecked devastation throughout the Balkans, however this resulted in the utter defeat of the roman army and was probably start of the process which led to the fall of the entire Western Roman Empire
Not the United States military, to any appreciable extent. It does have to worry about its internal laws though, in which restraint is assumed unless specifically excepted.
The police, on the other hand, can easily give in to the immediate fear of someone else shooting first: when they are not careful about not escalating, the worst thing that is likely to happen to their side is having to call in reinforcements and having some more bad press that won't add much to whatever they already have. If you want to put it in very crude terms, they are playing war in easy mode and there, many things that are very important to the military are completely irrelevant.