The answer is YES. The responses saying no are so wrong it isn't even funny. How do I know - I used to do tech work for a repair shop and saw all of the inner-goings.
A shop is a preferred shop because they will repair the car basically as cheaply as possible. They have to abide by certain things or they will lose their "certified repair shop" status. It is rigged.
You have a right to have parts that are OEM - original manufacturer parts. Anything else is an unknown even if supposed to be exactly the same.
You have a right to not have any part welded that or self repaired. Obviously this is inferior.
What should you do?
An example of an accident I was in is the easiest way.
I was involved in an accident where a young teenage girl was on her cell phone, going down a street, and decided to do a U-turn... well I was driving on the other side and she hit the front corner then proceeded to scrape (deeply) the whole side of the car.
So I took my car to one of their "preferred" shops. I got a quote of $3500. They were going to self-repair some thing, order knock off parts, and only paint the effected areas (hint - color fades, you can't just paint part of a car the original color).
Then I had to find a neutral shop - this is much harder than you think because 80% have tie-ins with insurance companies. It is a consumer scam. After I found one (small local shop) I told them - I want you to repair this like it is a show car. Boom quote was $6700 - almost double.
Insurance company told me to basically F off. Their repair shop said I was being unreasonable. I filed a small claims suit that day for $9500.
I received many calls from the insurance company. I would not budge so we went to court.
- Judge very quickly agreed that I had a right to have my car made new not look new. OEM parts, big paint job, the works. The insurance lawyer objected and judge asks - "Do you believe in the basic principles of supply and demand?" He says - "Yes" Judge says - "Well the OEM door is $1400 and your knock-off is $800. If they were the same the would be about the same price."
- I found statistics that reflected the additional repairs a car needed for being in a partial front-end and it was about 25% more than normal. I gave a low ball estimate for repairs for 5 years and came up with $800. Judge loved this but just gave me $400 - said she had no proof I would own the car over the life.
- The other $2000. Loss of value. The accident is put on your car's vehicle report and no matter how good of a job a shop does, it isn't how it came off the assembly line. This one was easy for me because I just went down to a local car dealer and said I was trading it in. Asked them how much less they were giving me because it was in an accident - they said $2000. I got this statement notarized and boom. (loss of value claims are now part of normal insurance dealings in many states but the insurance companies won't tell you).
I for some reason happen to be an accident magnet and honestly tired of the time it wastes of mine and then realizing how much money you spend because someone else hits you. The advice I would give anyone if they don't want to push insurance companies is that if you go to their preferred shop then go in and say "Since you are not using OEM parts and not repairing as new, I would like a lifetime warranty on everything involved because I plan on driving this car another 10 years." They won't give you the warranty and will often offer to just fix the car "right". I have been at shops and they were like "let me call the insurance company and see if we can get OEM parts for this" (scummy scam seriously).
Your case. You can't fix a cracked bumper. Get a new OEM bumper. Simply google OEM bumper for MAKE and MODEL and you will get the price easily. Also you have a right to just take their check and not fix it - just like the person who hit you had the right to not hit you. That is your money fix it if you want and has nothing to do with the insurance company. As I have mentioned in comments, you could just sue the person and the judge will give you a judgement of money and that said judge could care less if you bought cases of whiskey with the money or repair your car - DO NOT EVER LET THE INSURANCE COMPANY SEND YOUR CHECK TO A SHOP.
Note: I want to add a specific side note for the bumper. My friend got hit three times in their bumper - all slow rear ends kid you not. The first time they did not get it fixed and accepted about $400 from insurance company. The second time was less of a hit but a truck so it really scraped it due to height differences. The field service guy the insurance sent out saw the damage and then said let me check the padding under the bumper.
It was damaged/cracked which negates much of the damage control from the bumper unit. BUT this was probably done on the first accident. That insurance and their shop only was going to pay for the bumper cover not replacing the whole bumper unit. This was $400 compared to $1100 (including paint/install). The second guy was honest the first wasn't and neither was the shop.
So he got $1500 and still didn't fix it. Got hit again boom another $800 (they wouldn't pay out because the truck damage was noticeable. He finally got it fixed before selling it. Note this was before CARFAX stuff so not sure how that would be handled - would potential buyer in future see 3 accidents and 1 repair?