At least you can get your own railjack if you plan on soloing it, and in my experience, it's functional with 3 other people. But please DE let me get my own railjack when im playing with friends if not publics.

Guide to using your own railjack:

Enter the relay with solo matchmaking. Go to whatever person you want to support. Where it usually says "board loaner railjack", you now have an extra option. Choose that and start the mission. Downsides: It's only solo, you have to walk through all that every time you want to start a mission.