Captain reported inadvertently leaving the nose landing gear steering pin uninstalled prior to takeoff. Upon gear retraction; the crew received a gear unsafe indication and decided the best course of action would be to continue to the destination airport where better services were available.

Narrative

A Part 91 corporate flight was scheduled from ZZZ-ZZZ1-ZZZ. After arriving at the FBO at ZZZ1; the Line Department advised me they would need to tow the plane to a different location. I then unpinned the NLG steering for towing. After repositioning the plane about an hour later; it was scheduled to be fueled. Fueling was delayed due to a plane that was parked next to our Westwind. Fueling was then further delayed because the Line Department didn't want to fuel us until after the newly parked plane departed first. Meanwhile I was contacted that our Passengers were on their way and would be there in ten minutes. For the second time I approached the FBO Attendant and asked to be fueled immediately. The Line Department department delayed getting to the fuel truck quickly so for a third time I approached the front desk and canceled the fuel request. At about that time my passengers showed up and we loaded the plane for departure. Taxing out to the runway (Runway XX) was normal. After departing the runway; I requested the Landing Gear Up; immediately after Gear Selection Up; I noticed a Red Unsafe Gear Light and Pitot Static Amber Lights Illuminated. This is consistent with the Nose Gear Steering Pin Not Installed. I had the Second In-Command (SIC) contact the FBO via UNICOM to verify if the Line Department Re-Installed the Nose Steering Pin after repositioning the plane. Their reply was; they did not re-install the pin. I made the decision to continue on to our destination of ZZZ where I as very familiar with the airport and they had adequate ARFF equipment. I advised ZZZ approach of our possible lack of the nose landing gear not being down and locked and the possibility of loss of steering control and requested priority handling. Upon selecting Gear Down; the indication showed all gear positions to be down and locked. I made a normal landing but held the nose gear off the runway as long as possible. When the nose tires finally contacted the runway; everything was initially normal. Then as the plane de-accelerated; the nose gear developed a light to moderate shutter; then quit. Post flight revealed that the nose gear steering pin was not installed. I have since initiated a procedure to guard both start switches in the cockpit with a red device until verification by both crew members of the nose pin installation can be made before removing the device.