Again, from what you describe, I'm thinking your system is operating as best it can.
But to verify that, you need to establish the flow rates of all your fixtures. Use a known volume container and time how long it takes to fill up and convert that to gallons per minute. When you said, "when we turn the water on in the house or at the bib near the well pump the pressure drops, the pumps kicks on, the pressure rises close to 60psi, pump shuts off, pressure goes down and then the cycle repeats.", are you turning on one faucet in the house of multiple faucets? Figure the gallons per minute coming out of whatever faucets and/or bibb when the system cycles like you say is normal. Then determine the gallons per minute out of the garden hose at "full blast" that causes the well pump to come on but not build up the pressure to your set point.
I would expect the flow rate out of the garden hose at "full blast" to be significantly more than the flows out of your house fixtures or the hose bibb at the pump.
Below is a representation of a pump curve.
For argument's sake, let's double the "Head" numbers so it looks kinda like your pump and let's call those numbers PSI instead of Feet of Water. that green line is what the pump can do. At 15 GPM, the pressure the pump puts out is 60 PSI, and at all flows below that, it can put out a little more. So, when the pump is flowing anything up to 15 GPM, the pump will cycle based on the pressure switch settings. However, if the flow rate out of the pump is greater than 15 GPM, the pump can NEVER reach the set point of the pressure switch, and this will cause the pump to run continuously until the flow rate is slowed by throttling the garden hose, in your case, to something below 15 GPM. And at something like 21.5 GPM the pressure will only be like 40 PSI.
I will remind you that all these numbers are representative as you would need a pump curve for your particular pump. But this shows the concept of what I think is happening.
You may ask, "Why didn't the people I called in know this?" The answer is that they are paid to come out and perform a service. If I am right in my analysis which is based on what you describe, anything short of installing a new well pump will not fix your issue. But they can come out and tweak this and adjust that and get paid for doing that service.
If you really want your garden hose to have higher pressure, either only open it halfway or install an inline restriction that would reduce the garden hose flow rate and would cause your pump to cycle. But as Valveman and I have said, pumps love to run but hate to be turned on and off repeatedly and frequently. That is hard on the mechanical parts of the pump and the inrush of the electricity to get the pump turning from a dead stop isn't the best thing either.
What this representative pump curve doesn't show is the energy usage. As the work the pump does is moving water, the more water that is moved, the more energy/electricity the pump needs. So, when the pump is running at low flow rates, it uses less energy. This is where the Cycle Stop Valve comes into play. It keeps the pump running using less energy flowing only the water that is being used. Only when no water is being use is the pump turned off. A standard pressure switch system turns the pump on at a relatively high-rate flow to build the pressure, and then off when that pressure is met. A large pressure tank will reduce the cycling a bit, but the pump still pumps at a high rate to fill the tank and meet the pressure switch set point.
I hope this provides you a little more insight into "pumps, pressure switch, pressure tanks, etc."