How does tone-arm mass impact performance?

There is no answer for your question because there is more than one variable involved. For any particular cartridge there may be an optimum mass but there is no mass optimum for all cartridges. During the race toward higher and higher compliance which occurred during the 70s it was determined that one cartridge required an arm with negative mass for best performance. With the trend to lower compliance cartridges the mass of arms has tended to rise, although some of the older arms were very high mass. The arm/cartridge system is a very complex one and no single parameter will govern it's performance. The weight of the cartridge should have no bearing on the antiskate force, it is determined by the tracking force , not the weight of the cartridge. The compliance of the cartridge determines the mass of the arm, low compliance-high mass and vice versa. If you get this wrong the resonance frequency of the combination will be either too high or too low. See sites like Vinyl Engine of Audio Asylum for fuller explanations.

Excuse typo, Vinyl Engine OR Audio Asylum

The relationship between the arm and the suspension of the cartridge can be modeled like a spring. The springier, more compliant, the suspension, the less mass is desired in the arm to bring the total resonance of the system to the optimal point. Likewise, if the suspension is stiffer, more mass is required in the arm to keep the total system resonance at an appropriate frequency.

Interestingly, the weight of the arm is not of importance. What is importance is the effective mass, which takes into account the distance that the mass is from the pivot point. An extra gram at the headshell increases effective mass much more than a gram near the pivot point.

If the resonance of the system is too high it can be excited by low freqency tones within the music. If it is too low, warps will cause the resonance to be excited.

In your example, yes a one ton arm and one ton counterweight could play a record, provided that the compliance of the cartridge was suffienciently low. This is kind of theoretical though.

Commercial arms vary from somewhere around 6-7 grams effective mass all of the way up to 30 grams or so. Most medium mass arms, like the Rega, Audioquest/Jelco, Technics, etc. fall into the 9-10 gram range.