tone arm drifts towards the outside when lowering

Is your turntable level?

An issue here. Quoting VPI , without anti-skating , and using a pivoted arm with an offset head, "Putting a stylus down on a flat grooveless record will cause the arm to shoot in toward the center of the record." If you use anti-skating at all (some don't), you want to COUNTERACT this tendency, not bolster it. Thus the OP's situation is better than the opposite, but it sounds like he has a bit too much anti-skating, for whatever reason.

check to make sure the contact points between the arm and lift platform are free of dust and oil.

Check your antiskating by carefully viewing the cartridge head-on while a record is being played. Viewed from that perspective, the cantilever should be at the same left vs. right angle relative to the cartridge body that it is at when the stylus is lifted off of the record (which for most properly designed and manufactured cartridges will be straight ahead).

In my experience, most of the guidelines for setting anti-skating force which may be offered by tonearm or turntable manufacturers result in too much force being applied, and I have typically found that about 2/3 of the recommended setting gives the right results.

Regards,
-- Al

Assuming a tonearm is unconstrained in the lateral plain during cueing (everything level, dust- and dirt-free, low friction bearings in good shape and properly adjusted, tonearm wire twist and other lateral forces eliminated, little friction between armtube and cueing support), then any amount of anti-skating applied during cueing will move the arm outward. That's what anti-skating devices do, push or pull the arm outward.

Actual behavior varies from one arm design to another. For example, outward cueing drift is typical with Rega arms and their clones, because (1) the anti-skate device applies some force even if set to "zero" and (2) that force is always on. OTOH, cueing drift on a TriPlanar is avoidable. Aside from the option of making anti-skate truly "zero", the anti-skate device is readily adjusted so that outward force is not applied until the stylus touches down.

Other designs may vary in other ways. Check all the factors posted above. If all seems okay then investigate the anti-skating functioning of your arm. Its design may or may not allow you to mitigate outward cueing drift.