I inevitably dissuade visitors to the Azores from day tripping to any islands. Why? The weather is always an unknown factor. Ferries and airplanes get delayed and cancelled more times than you think throughout the year, and even in the summer. I can assure you that there will not be any lodging options for a cancelled ferry or airplane if you get stuck overnight in the summer nights, unless you are in Sao Miguel or Terceira. Also, the other seven islands have very few taxis available, and they’re not easily accessible. Unless you reserve a rental car many months before your summer plans, you will be without one.
The ferry to Sao Jorge, from either Pico or Faial, is at least 60-90 minutes, each way, in good weather. Don't expect taxis to be waiting at any port, except in Ponta Delgada, Sao Miguel. I once took a ferry from Madalena, Pico, to Horta, Faial and zero taxis were waiting at the port. Some passengers needed to quickly get to the airport to catch a flight, but there was not much anyone in the very long line could do except to keep waiting. Fortunately, I was heading to the airport to pick up my rental car, so I shared my taxi (after waiting 30 minutes) with another passenger.
You are correct about getting to Faja de Santo Cristo only by foot or quad. The quad option is next to Ermida de Nossa Senhora de Lourdes in Faja dos Cubres, at the cafe. I believe there is no way to reserve a ride (not 100% sure), and highly doubt they are rentable. There are not that many quads to shuttle anyone from this spot to Faja do Santo Cristo, from the times I have been there. I was tempted to hike the route, but felt it was not worth the time or effort. There are parking spaces and bathrooms before you enter the forest up the hill, and also next to the ermida.
I have only hiked a handful of routes in the Azores. Every one took longer and entailed more effort than what is described. The best hiking website is: https://trails.visitazores.com/en
My guess for this specific hiking route is probably it is not that scenic compared to many more places on this island, even those accessible by car. The start of the trail at Serra do Topo is nowhere near as stunning as countless other areas of Sao Jorge. There is a miradouro there, but the view is just ok. This route is not one that starts and ends at the same spot, complicating your plan. I prefer trails that get me back to my car.
Sorry to give you advice that might not be what you were expecting. If you end up staying a few days in Sao Jorge with a rental car, then you can accomplish what you are thinking.
