Yes, but evil old school bank likely would not process 'card not present' transactions at all.
Really, I don't see what this contributes to the discussion, I would assume that if the OP was able to sign up for their own merchant account they would have done so. To process six figure sums on someone else's merchant account is usually for a reason other than convenience because the fees are high enough that you will want to cut out that particular middle man as soon as possible.
And yes, it's a better solution than Stripe, assuming you are not a high risk merchant, in which case banks won't touch you.