Because of a recovering knee injury, I have been advised not to do full squats yet, but to build of some strength doing partials, then box squats. You say the deadlift with the trap bar resembles a partial squat more than a deadlift, you think this is a good exercise to use?
Who gave you that advice? What was your knee injury? Have you seen a doctor of sports medicine or a physiotherapist?
If you got that advice from a GP (general practitioner AKA family doctor) it would be good to get a second opinion. I'm not a doctor, and I'm not going to contradict what a doctor says, nor can I diagnose over the internet...but I do know that GPs, while knowing many things, don't usually have the best understanding of sports medicine, injuries, rehab...and I've been told as much by one of the guys I lift with who is a doctor.
One issue with partial movements (when doing just the top part) is they allow more weight to be used, but only work a limited part of the ROM. Altogether increasing the risk of injury. This is especially true in squats, or in this case, trap bar deadlifts. Which is why they should have a limited role, if any. Which should be even more true if you're recovering from a knee injury. Keep in mind that strength is specific to joint angle, so the strength you gain from doing partials will carryover very little to full squats...which means it makes little sense, to me, that someone would use partial squats to rehab and progress back to doing full squats.
If the advice given to you by the doctor is sound, I'd suggest prowler work or sled drags (car pushes if neither of those are an option) and step-ups over trap bar deadlifts or partial squats. Trap bar deadlifts could still be done, but as assistance or for variety. And of course regular deadlifts and posterior chain work should be done as well. You could also consider the use of bands or chains at some point, to reduce the load at the bottom of the squat,
