In general the answer is no, you do not need the thumb to be curled up to bend. However, this depends on the gauge string, set up of the guitar, and your experience.
On the flip side I think the "standard" or "correct" way to hold the neck is a bit misunderstood. Physics can help understand what is going on. First of all note that this so called proper way originates in classical guitar. These guitars have very thick necks, thick in both directions transverse to the length. Most normal sized people could not get their thumb around it and play.
Many people think the thumb is used to squeeze the string down when playing. This is in fact not true. When held in correct position the finger board should be a vertical plane, even leaning forward a bit with the finger board pointing to the floor (at a very slight angle). The fingers should not require any help from the thumb and many experienced players can play with the thumb completely detached from the neck. When set up correctly (i.e. with good action) the amount of force required to press the string down to the fret and make a good sound is very small. The same holds for upright bass. With correct posture and a good set up it should feel effortless.
One thing is that is correct regardless of how you hold it, you need to be mobile. Squeezing the neck with the thumb in back or curled over the edge will prevent you from being able to move easily. This is not a good way to play.
That being said, when it comes to bending a lot more force is required to bend a string in the plane of the finger board than is required to press it down. Exactly how much more depends on string gauge and set up but more is required. Personally I am able to bend with my thumb in the classically correct position (behind) but I don't force myself in this habit. If I'm bending more than a minor third I do need the extra help and there is not enough friction between the neck and my thumb to prevent the whole neck from popping out of my hand. But I would caution that I do not squeeze the neck, I just place the thumb over the edge to prevent the neck from leaving my hand on extreme bends.
If you are going to use that technique you will want to keep the hand loose to maintain mobility.