Why did this progressive insurance company not take my offer? ?

Because they knew they had a sucker. (You)

You had every right to have that money (The entire $4500.00 check plus some rental costs) be made out to you and to you only. When another company pays, you do NOT have to repair your car. When they saw that you were willing to settle for less; only $3,000., they knew they had a sucker so they rubbed their hands together and said "Lets offer this sap just $1800.00.

I suggest that you make it your business to understand automobile insurance. At least the basics.

There is no way to have a legal opinion based on this sketchy information.

You are leaving important details out. Im guessing a reasonable cost for the repair was about $1800.  You don't get rental car insurance.

You're leaving out too many details.

Ask the adjuster. Maybe your car is only worth 1800.

Did they pay the $6K or did you? and you are waiting for them to reimburse you.

Without knowing that, we can't answer your actual question. 

First off, a car rental is not included in car repairs.  Car repairs were just $4500.  IF you are entitled to a rental car, the insurance company would pay for that - separately from the garage that made the repairs to your car.  You lost out because of how you handled it.  You were offered $1800.  That means the insurance agent assessed the damages at $1800.  You probably should have either taken the car to a garage that doesn't charge through the roof or had them talk to the garage people themselves to argue the price with them.  Since you chose to handle it yourself, you were offered what they figured the damages were $1800 and you chose to independently have your car repaired for $4500.  That was your choice.  You took it out of the hands of the insurance company.  And, it looks like your insurance policy did not cover a rental car, when needed.  If that is what you wanted, you needed to make sure you asked for it and were charged for it.  Then, you would be covered for it.  Lastly, you said you wanted 3k.  That is $1200 over what they assessed the damages to be, so you are not going to get 3k.  You refused the 1800 in favor of 3k that you were hoping for.  You need to realize that they are not going to pay more than the value of the services rendered, which they valued at $1800.  It is far too late to argue price - after the fact.  It's all a done deal at this point.  Next time, if you run into trouble and you don't know what to do about it, hire a lawyer.  He will set you straight and get you what you are entitled to but, in this case, it was JUST $1800, the assessed value of the damages.