I can only speak from a practical view point and not a theoretical one. IMHO what we are trying to achieve is a safe practical approach & landing. I assume whether you are flying Boeing or Airbus you would elect to fly a circling approach, where the weather was near minima, using autopilot. It doesn't matter if you fly the initial approach via an LS or LOC; you round up the circling MDA - oo's and set it in the Altitude capture window. The a/c will 'start to level off' at a suitable height above that setting and then level off AT it. It should not bust it and there will be no sink and you will be above MDA.
Downwind, from abeam threshold, in a Cat C a/c I've always used 3sec/100' above threshold as datum and deducted 1sec/2kts tail wind. i.e. half the tailwind. A rate 1 level base turn. If you've made the correct spacing and turn radius you should roll out very close to centre line and on a 3degree GP. When you are about 90degrees to go you will be entering the +/-30 degree cone to the centre line where you can start a visual descent. You are now 'on finals' and terrain serration is Mk.1 eyeball.
Like I said, it seems to have worked for me over the years. There are one or 2 places where I question the use of the term circling. e.g. LEXJ RW29 Santander. The 'circling MDA is 2490' agl (2510' QNH) and the vis required is 2400m. Good game if you can't descend until on finals. I asked the chappie who writes the Airfield Briefs about this, as no mention/guidance was given about what to do. The SOP would be to set 2600' in MCP. You would not be visual at MDA with only 2400m. (Even the CAT D is 3600m.) These are standard category distances, make no sense and bear no resemblance to MDA, unlike a NPA CDA where the required Vis is related to the distance from RWY at MDA.
So there you are at 2600' and decide to fly level to the MAPT. You the see the required visual reference of the runway environment and commence circling for 75secs -1/2 tail wind. That will put you very close to the 4.2nm circle of safety. You now turn finals at 2600' and 4nm. That is twice the normal height and will need 1400-1600fpm just to get to the runway = unstable at 1000' & 500', unless you do 2000fpm PDQ. Equally, with 2400m vis you'd lose sight of the runway very soon after passing the threshold. All in all it is No Way a circling approach. It is a cloud break procedure for an extended visual circuit. The circuit takes you over the city, hence the height. All very confusing, but like I said, no guidance given and I'd expect total confusion to anyone who tried the conventional profile.
It sounds like Dalaman is similar. Not needed to do it in anger. Guess I missed something. Not one for the trained monkeys. Given that vis is an approach ban parameter, and circling is fraught with danger and vulnerable to total screw ups, IMHO it is not professional for the authorities to label such an approach as 'circling'. There will be places where 'get-home-itis' will sucker guys into scratching around on circling approaches in dodgy weather and hurtling at the runway in an un-stable manner. Not good.