Clogged vent is an issue for different reasons, and should be checked and fixed if it really is clogged. But that condition should only cause the temp inside the dryer to rise until its normal thermostat shuts off the heating element, then turns it on again later when the temp drops. This repeat sequence caused by inadequate air flow (and hence removal of heat and water vapour) certainly can make the total drying time for any load too long and thus waste electrical energy and increase your costs. But it should not cause any electrical overload and breaker tripping unless there is a fault in the dryer's components.
Regarding the aluminum wiring, IF it is an issue (it usually is not, but is possible), the real problem is that there is oxidation of the aluminum at contact points, making excessive resistance there. That means generation of heat at the contact point and a reduced voltage going to the load. The result of that is reduced max current flow thought the load, so the impact on the breaker is to REDUCE the current, and therfore reduce the probability of tripping. The practial hazard of this is the HEAT generated at a connection point in some small junction box, and whether that can damage the wiring and insulation. It is NOT a higher load on the circuit. Now, in a rare case there MIGHT actually be damage to wires somewhere in the circuit at a junction point (HIGHLY unlikely inside the cable), so you could open and inspect every junction point box for visual hints of damage. But unless there is major damage causing significant leakage to Ground, that will not affect current load through the breaker.
A typical stand-alone dryer fed with 220 VAC has a 4500 W heating element, so that's just over 20 A. Add the ¼ HP motor that operates on 110 V, and that's another 5A or so. Allowing for motor starting inrush current, most estimate that max load to be about 28 A. A common rule of thumb is to say the current limiter (in this case, the breaker) and wiring should be chosen to operate in normal circunstances at no more than 80% of its rating. So such a dryer often is fed from a 35A breaker of fuse: 80% of 35 A is 28A. You have a stacking system that, as you say, is designed to use less power. BUT the wall outlet is designed to feed BOTH the washer and dryer simultaneously. Your dryer very likely has less than 28 A max load, but the washer adds another 5 to 6 A for its motor, so we're up in the 30 A max load combined. That is why the supply line and breaker are chosen for 40 A. Now, your post indicates that this breaker tripping happens very often when ONLY the dryer is running, and the breaker is rated well over the proper load for that device. As Greenman said above, something is seriously wrong!
You say the load was measured as 14 A. I suggest you need to get that re-measured and monitored while running normally, watching carefully for any sudden change just before the breaker trips. It will be a large spike to trip a 40 A Breaker! Start by measuring current flow right at the output of the breaker, which is what the breaker responds to. This is a bit tricky because you have TWO Hot lines to check. IF this indicates that the actual current flow out of the breaker DOES spike and cause the trip, then you need to follow the circuit further down to find where. The next test point would be the lines right at the dryer, to see of the current spike is caused by something in the dryer. On the other hand, if there is NO current spike but the beaker trips anyway, maybe the breaker is faulty.
I just thought of something I ran into recently. I understand that in many jurisdictions now, the current-protection device being installed (often because of new rules) are NOT the classic breakers or even the newer Ground Fault Circuit Interruptors (GFCI's). A Breaker trips due solely to high currents for a sustained period of a few seconds (a bit faster for a real short circuit). A GFCI trips for that AND for a detected imbalance of current flowing out on a Hot line vs current returning on the Neutral line (indicates leakage current out of the intended load). The NEW devices coming into use are called Arc Fault Circuit Interrupters (AFCI's), and they will trip due to current overloads over a few seconds, AND for a sudden high current of very short duration that indicates a brief short circuit caused by a flash arc somewhere along the circuit. Such events can result from deteriorated insulation, improper conductor spacing, debris bridging between wires, or other items. These situations can cause recurring arcs that may ignite a fire and hence are dangerous, even though they last for such a short time that they cannot cause the wires themselves to overheat and fail. As I understand the situation, these new protection devices are somewhat controversial becasue many tend to trip out too often due to normal and acceptable short-duration current surges in equipment that is NOT faulty, but create brief arcs and sparks. That might be a motor or a switch or relay, for example. So, examine carefully exactly what type of "Breaker" you have in your panel. If you replaced an old faulty breaker that was tripping too soon with a new AFCI type of breaker that is prone to tripping too easily, that MIGHT make your situation appear to NOT have changed with the replacement breaker.
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