I've had several missions ( not in quickplay) start like two or three quest steps from the beginning for missions I have never done. Sometimes I'm loaded in so far from my allies that I hit another loading screen to get to them due to the auto catch up mechanic. Some of these missions have been critical story missions where I'll miss dialogue that set it up the mission so the context is completely lost. I have also had this happen with contracts ( which I know can be shorter) and again lose the context of the mission leaving me at a complete loss as to why I'm doing whatever it is I'm being prompted to do in a mission.
I suggest having all players load in and then have the mission start. So, all players spawn in and start the mission together. That way nobody feels lost doing a mission they have never done.
I would expect loading into a ongoing mission for quickplay or even a stronghold but not for regular match made missions. Let me know if you need me to elabora...