I don't really understand the term "clamping". Does that mean it will try to assume the reference voltage but the actual voltage has to come from somewhere else? And if it can't get that voltage, it won't hit the reference? In that case, the voltage between the output pin and the adjust pin would be lower than 1.25V until the current limit is reached. Only then would the regulator start regulating at all.
Yes, perhaps "clamping" is not the best term, something like 'reference turn-on voltage' might be better, but you've managed to 'decode' the rest of it correctly - the regulator won't start doing any regulating until the voltage between the Adj and output pins reaches 1.25V (it just passes the input to the output, minus the dropout voltage of the regulator, around 2V drop depending on load current). Yes, you've got it, the actual current to the Adj pin must come from something external, either the shunt resistor in the current source case or a resistive divider to ground in the case of a standard voltage regulator. If you simply connected the Adj pin to ground then the regulator would output 1.25V.
Don't give up on the switched output current if you're still trying to keep things very simple (ie not trying the 723 yet) then it will still probably be sufficient in most cases - just limiting the current to protect your circuit. You could add extra positions to go down below 100mA, eg 50mA 20mA 10mA (1:2:5 ratio tends to work best), low enough for LEDs for example. Note that the LM317 needs to pass at least 5mA typ. to power its internal circuit, but you can just add a resistor from the output to ground if necessary.
Glad it was of some help.