punctuality and safety are not mutually exclusive.
certainly rushing checklists or screwing up is not part of any equation and there are ways to shave time quite safely...however some do not know how.
have checklists done early, so you do not rush them.
and being fast below 10,000' isn't so much for the birds but for collision avoidance.
asking ATC for direct is just fine...period...they are there for you, not the other way 'round.
there are ways to plan ahead...be ahead and make up a bit of time...one thing to understand...if you make up 3 minutes in the cruise area, you might be ahead of a dozen planes in the terminal area.
you might also be three minutes ahead of a storm which would cause you to divert, refuel, and end up more tired by the time you get to original destination...each additional landing/takeoff exposes you to more hazard.
you said you can't change arrival time by more than a couple of minutes, so why try?
like I said...if you are three minutes early, you are 'head of the pack', beat a storm, allow for a connection to be made and so much more.
so, nothing wrong...except things like taxiing too fast, breaking regulations, not doing checklists...but those things can happen while ahead of schedule too.
have things ready well in advance...get the clearance after the shutdown check is complete...you might find out about a gatehold early!
call in and request a quick turn from ground personnel.
and isn't it funny when a lightning detector dictates ground folk can't come out and help you park?
safety first...but there are ways of moving along quite safely and even making safety even better.