Safelite and the state of Connecticut have settled out of court.

It says that the first part of the law can still be enforced (although it isn't) and only the part that required Safelite to name a competitor alternative cannot be enforced. It also said that Safelite had to pay their own expenses. Here is what is left to enforce:

Sec. 38a-354. (Formerly Sec. 38-175u). Automobile appraisers and insurers prohibited from requiring where repairs should be made or making certain statements. Notice required on appraisals or estimates. (a) No automobile physical damage appraiser shall require that appraisals or repairs should or should not be made in a specified facility or repair shop or shops.

(b) No insurance company doing business in this state, or agent or adjuster for such company shall (1) require any insured to use a specific person for the provision of automobile physical damage repairs, automobile glass replacement, glass repair service or glass products, or (2) state that choosing a facility other than a motor vehicle repair shop participating in a motor vehicle repair program established by such company will result in delays in repairing the motor vehicle or a lack of guarantee for repair work.

(c) Any appraisal or estimate for a motor vehicle physical damage claim written on behalf of an insurer shall include the following notice, printed in not less than ten-point boldface type:

NOTICE:

YOU HAVE THE RIGHT TO CHOOSE THE LICENSED REPAIR SHOP WHERE THE DAMAGE TO YOUR MOTOR VEHICLE WILL BE REPAIRED.

For all the legal scholars out there, if it violates a company's right to free commercial speech to compel it to communicate the name of a competitor to someone, then why isn't also a 1st Amendment violation for an insurance company to compel a repair facility to communicate proprietary business information to its competitor (Safelite)?

The needle on this meter says you are asking the correct question.

Seems like an idea that might work is legislation that stops ALL marketing
at the time of reporting a claim.

Not a legal expert but it is my understanding that the 1st Amendment protects citizens from government laws that restrict or compel speech and since the state of CT is a government its law was subject to review.

Since insurance companies are not "governments" the communications that they compel are not subject to review at least for violating the 1st Amendment.
However, there is the Consent Decree of 1963 which defines steering and other laws regarding price fixing, boycotting, collusion, ethics, fair competition, deceptive business practices etc.

Safelite rightly argued that to be compelled to mention a competitor would make people think something was wrong with Safelite. When they made that argument, Safelite showed how two faced or hypocritical that they are.
Safelite knowingly colludes with an insurance company every time they contract with one. They and the insurance company knowingly agree to put Safelite in a position to lord over their competition in a biased way.
They and the insurance company knowingly agree to allow Safelite to form a network that results in price fixing. They and the insurance company knowingly enter into an agreement that charges Safelite to enforce or at least, attempt to boycott repair facilities that will not participate in the price fixing network. Safelite and their contracted insurance companies knowingly violate the Consent Decree of 1963 whenever they sign the contract.

In other words, the Safelite business model is corrupt. Their business model puts them into a position to do exactly what they argued was wrong to their own competitors. Safelite can no longer argue that their claims administration division is not a competitor to a repair facility because three Federal judges have ruled otherwise. In fact, any TPA that uses the GAI scheme is a competitor to a repair facility.
Any "network" that includes setting pricing violates the price fixing antitrust law.

Sometimes the truth hurts.....

SO if we take this 1 more step....any and all shops who AGREE to network pricing are "PRICE FIXING". Any 2 parties or more attempting to "set pricing" are engaging in price fixing. So when any shops agree to network pricing being set by outside forces that collaborate to determine a set price that others have to follow they are engaging in price fixing.... please show me where I am wrong.

It is in fact price fixing. Safelite is taking a specific area and setting a price (% off Nags and labor hours) to every job completed. The shop has to agree to the FIXED PRICE in order to do the work through the network (Safelite).

Price fixing 100%. Oh, and if the repair shop charges more, they short pay to the FIXED PRICE.

Anyone that cannot see this is blind.