I can be wrong, but I thought patrol work was ...... well.... patrolling. not laying in the harbour until you are needed.
I can be wrong, but I think you don't have much of an idea of what's actually involved in the mission profile of a patrol vessel.
Patrolling
really means establishing a presence. This means you have to be
present - as implied by the word "presence" itself. This usually means a single ship covering a lot of open water, because those are usually the resources on hand (even a resource rich nation like the United States has resource restrictions on how many boats it can put in the water, between funding other defense/police needs, funding domestic needs, and not wasting taxpayer's money). This means, as converse as it may seem, a
slow cruising speed - you want to maximize your time out on patrol, and most importantly, not waste a lot of fuel.
Look at it this way - your local police, yes, speed to the scene of a response call, but when they're doing patrol work, they cruise around (this is why their cars are called -shocking! -
cruisers) pretty slowly, driving around neighborhoods and other "beat" areas at a relative crawl, especially when a lack of traffic lets them get away with it. Not all of it is to save fuel either - you're establishing a visual presence; you're letting people around you know that you're obviously looking for any trouble and deal with it. That's what a large part of policing is about.
And it's no different for a patrol vessel, either. The vast majority of patrol vessels are looking out for civilian boats - anybody who might be a drug runner, smuggler, or even as seemingly banal as illegal fishing (this last part probably makes up at least 80% of all oceanic patrol targets). This is why its stupid to put anything bigger than at the very most 76mm automatics - what you're really looking for is to scare the crap out of some careless fisherman or poorly equipped smuggler with the
threat of killing him and sinking his boat, so that he can listen to you and either turn around or surrender. If he's say, an actual military force arriving in a missile destroyer - well that's what your navy and your own destroyers are for!
Anyway, most of what a patrol vessel does is be visible - most fishermen are going to tend to their designated areas, and most smugglers aren't even going to think about doing their business as long as they know an armed presence is nearby. So you need a ship with deployable sustainability. If they decide to run - well, you can still have a high dash speed, but this is also why the USCG has Dassault Falcon 20s in their air fleet. It's hard a 50 knot or even 75 knot speedboat to outrace a bizjet flying at Mach .80.
Note that the USN retired the Pegasus class FACs into the USCG to give the USCG the exact type of fast response that an axe-bow cutter would be designed for. The USCG couldn't get rid of them fast enough. They lacked the staying power that the USCG really needed. Also, the USCG does have fast response interceptors. These are usually RHIB boats that can be deployed from larger vessels when needed.