Can this really happen where a speed difference of 30-40 knots can be undetected and put the crew into a situation where normal law refuses you the ability to manage the pitch attitude?
Affirm. If two ADRs indicate the same speed (despite being in error) then system logic will accept them and reject the 3rd one.
If you have totally blocked pitots and are still climbing, then the indicated airspeed will increase (due lower static pressure) and if overspeed protection activates then the aircraft will pitch up uncontrollably in Normal Law.
Don't be afraid to put the aircraft into ALT LAW to regain control (unless you are like the crew of AF447 or QZ 8501).