For 99% of the straight-in ILS Category 1 approaches, the localizer and glideslope will lead you precisely to the touchdown zone, so most manufacturers/operators trust it well enough to leave the FDs on until landing. Some airplanes remove them at 50ft (or similar) anyway. The cases when you have troubles following FD on a Cat 1 approach below minimum are rare, but still might happen, like the Singapore B777 incident in Munich (
Incident: Singapore B773 at Munich on Nov 3rd 2011, runway excursion).
As far as the non-precision approaches go, you have an enormous amount of variety that goes with them: NDB, VOR, RNAV, offset, straight-in, ... And even a straight-in non-precision approach will mostly bring you from one or the other direction, non of which is completely straight in (unless flown with FMS, assisted by GPS). Not to mention there are multiple lateral and vertical guidance modes one can use when flying NPA: HDG, TRACK, (L)NAV, VOR/LOC (lateral), V/S, FPA, VNAV (vertical). Some of them may be appropriate to use below the minimum (VNAV on some Boeings or FINAL APP on Airbus), but it is much more simple to use the "one for all" rule and just switch off (or recycle on some aircrat) the FDs and fly the rest of the approach visually - and anyway, to continue the approach beyond minimums, you should have sufficient visual references to begin with.