Has there been any attempt to use an underground cable system, a la a cable car, to tow aircraft along taxiways?
Way to go on thinking outside of the box. Actually, it might work. Although the logistics of it are above my pay-grade. Maybe, some smart group out there could make it work.
But, your question doesn’t ask for ways to make it works. It specifically asked for reasons it might not works. So, here it goes:

cable car feasibility over all does not have a great track record. If we were to make a Ben Franklin style decision graph of cable car systems that worked versus those that proved ineffective, Wikipedia lists a lot more defunct systems than it does working systems. All of which run on tracks.
If you get more than two runways at the same airport, the taxiway system become increasingly complex. An aircraft would need an alternate means of locomotion when switching from one runway/taxiway to another. The use of one continuous loop cable would be rendered impossible.
As Ron Beyer has pointed out, departing aircraft would still need time to warm up their engines prior to take off. Currently, that time is spent taxiing. Arriving aircraft would still need to power climate control systems. It would be a cheaper and more efficient use of time and resources to design a system that eliminates wait time between start up and takeoff. Keeping the traffic flow moving at all times without having to retrofit airports or entire airline fleets would benefit all.
Most cable car systems operate with cable cars of lighter weight and similar mass to each other compared to commercial aircraft. How would airports be able to account for the sheer mass differences between a single pilot C208 after a long cross country versus an A380 taking off at MTOW.
Power failure.
Power in night time IMC with four or five airliners about to break out at minimums for GPS approaches on parallel runways.
In the case of a constantly moving cable, an aircraft the size of an A320 could cause a lot of damage if it were not able to unhook itself from the cable at the appropriate time,
According to Wikipedia, the efficiency of a large cable car system is questionable. “Approximately 95% of the tractive effort in the San Francisco system is expended in simply moving the four cables at 9.5 miles per hour.“