Playing without the rest of the band? I'd walk away from it. It's a two way thing. While you want to know how the prospective player works, they also want to know how the rest of the band works. All very well trying out someone on a one to one basis, where they have little to blend in with or work off, but see it from their side too.
Some of the bands I've 'joined' and gigged with, there's not even been a rehearsal, let alone an audition. but they're reading bands, so the situation is maybe far different from yours. Others have had numerous rehearsals, including the introductory one - with the full band. Sometimes there's been a list of stuff a week before, sometimes I don't know until the practice.
And why do you think your view matters more than the other players? 'Cos it's your band? That's all very well, but may backfire when the drummer or another member decides he doesn't get on with the new member, either musically or personality wise. Then it's look for a new drummer time!
There are, as Carl indicates, no 'normal' practice with prospective new members, mainly as there are so many different types of bands, groups, orchestras, etc, with huge differences in how and what they play, and how they rehearse. Although for a mainstream soul band, with maybe 8 or so players, I'd always go for a full band first play - horn players would need to have the others playing at the same time anyway.