"Do my best" and Project Management

The idiom "doing our best" has become a bit of a cliche and essentially meaningless. Everyone claims to having done their best from star athletes, politicians and now project managers. Typically we use this term when we fail to deliver - I did my best. The question persists - did we really do our best?
In a team environment not only do the individuals have to perform but the team as a unit has to deliver. Every one can operate at 100% (best) yet the team fail due to internal forces; lack of communication, collaboration, integration as well as external forces; politics, weather, resource shortages, market conditions. Individuals and teams do not "do their best" unless driven to it through incentive, motivation, organization, leadership and yes, luck.
To me "doing out best" is admitting failure not only for past efforts (failure to deliver) but future efforts (we can't do any better [as we have done our best already]). I think the better idiom is - "we (I) will do better next time!". It's okay to admit weakness as long as you strive for improvement.

You are assuming here that "do our best" is not the same as "do what is required". "Do what is required" may be same, more or less than we are capable of.
If one does more than what is required we could be accused of applying too much effort.
If one does less than what is required one could be accused of insufficient effort or even failure to perform.
I would suggest that one applies sufficient effort to achieve what is required - no more, no less.
Quality is measured against requirements, not some concept of "what is best".
I see this problem with provision of professional services where the service providers ignore the clients needs and budgets (scope, cost and time) in order to provide, what is in their mind, the "best" solution. The client may see that delivering on the requirements is the solution that "best" meets their needs.
As I indicated earlier I see "we will do our best" as a set up for failure. If I am interviewing for a position (be it service provider or staff) when I ask "Can you do the job?" an answer such as "I'll do my best" suggest they are not confident in their ability to deliver.
By now you will understand that I do not like the idiom "doing our best".

I'd say Project management is doing our best to:
-Stay informed on what we require for success (Requirements Management),
-Making sure that our budget will be enough for completion.
-Escalating issues and risks promptly.
-Helping our upper management have a high level view of our objectives, timeline, and alignment to company strategy.

You are assuming here that "do our best" is not the same as "do what is required". "Do what is required" may be same, more or less than we are capable of.
If one does more than what is required we could be accused of applying too much effort.
If one does less than what is required one could be accused of insufficient effort or even failure to perform.
I would suggest that one applies sufficient effort to achieve what is required - no more, no less.
Quality is measured against requirements, not some concept of "what is best".
I see this problem with provision of professional services where the service providers ignore the clients needs and budgets (scope, cost and time) in order to provide, what is in their mind, the "best" solution. The client may see that delivering on the requirements is the solution that "best" meets their needs.
As I indicated earlier I see "we will do our best" as a set up for failure. If I am interviewing for a position (be it service provider or staff) when I ask "Can you do the job?" an answer such as "I'll do my best" suggest they are not confident in their ability to deliver.
By now you will understand that I do not like the idiom "doing our best".

I'd put "Doing one's best" in the same bucket as "trying". As Master Yoda said, "Do or do not, there is no try."