If the family member (applicant 1) is unlikely to receive mortgage approval with one lender, they will be assessed on the same criteria with another lender.
If a joint borrower (applicant 2) is needed for approval, regardless of whether applicant 1 is in practice able to meet repayments on their own, the lender will not let the applicant 2 just remove themselves from the mortgage. It isn't a case of just getting initial approval and then being able to adjust the terms. The mortgage is on the lenders terms and conditions and unless applicant 1's financial status improves sufficiently to be assessed as a sole applicant they won't allow the terms of the mortgage be adjusted.
Regarding the title deeds, on the initial application the lender may allow for joint borrowers sole title.