"Do rig counts tell us much about oil and natural gas supplies anymore?"

No.
That's why we quit posting them.
From Platts:

Consider the facts: Lower-48 gas production averaged at 72.1 Bcf/d in January, close to all-time highs, according to the US Energy Information Administration. Gas rigs, meanwhile, were at 14-year lows, averaging at 434 for the same month.

Simple answer to the question: Not if you’re trying to gauge supply, they’re not.
The follow-up question, as we say in journalist-speak, is far more interesting.
Why don’t rig counts matter anymore?

First off, drilling efficiencies are startling.

Drilling times per well have about halved compared to two years ago. The fastest time an Eagle Ford well has been drilled, as per an operator there, is three days. Three days.
Longer laterals, multiple frack stages…all these have combined to pull out more supplies in a faster amount of time, meaning less rigs are needed to produce more, more, more.

Another more elusive reason is what ISI Group’s Oil & Gas research crew recently termed the longer “spud-to-sales” time.

In a Tuesday analyst note, ISI noted nearly two decades ago, the ability to get supplies to market was a much easier affair. “Drill an 8,000 foot dry gas vertical, bring it on-stream and connect it to sales that week. With a short time between ‘spud to sales’ following the rig count was an effective way to gauge supply additions.”

That is clearly no longer the case.

In Pennsylvania, more than 1,000 wells have been drilled, but are waiting to be completed because the necessary takeaway capacity isn’t ready. Some producers have reduced their gas rig counts in the state to a grand total of one and still see their overall production levels grow year-over-year, a stark indicator of how prolific some of those wells are....MORE

Attentive readers probnably caught this tidbit:
November 2012

...We are seeing a battle between the rapid declines of shale-gas wells older than a year vs. hundreds (it might be 1300) of wells that have been completed but not hooked into the distribution system.
Further complicating analysis is the fact that almost a third of the country's nuclear fleet is down, refueling, Hurricane Sandy, San Onofre etc. putting more of the burden on the peaker gas generators which have been substituting for base capacity....