@Farnaz Z. so... it's complicated. While it appears to be a clear cut case of negligence, getting financial relief can be more challenging. Clearly you are unhappy. You have called and they have stonewalled according to your accounting.
If this tenant, in fact, had no previous evictions, would your opinion of the PM change? Eviction records are challenging as there is no clearing house for them like there is with say credit reports. What was their credit like? I find it very difficult to believe that a tenant had 4 prior evictions and you got them out in 3 months. With that many prior evictions, your PM was very on top of it. Most people with that many evictions never pay (they are professional scammers), and then use the court system to stall and stretch out the whole process. Unless you ran a full credit, criminal, and eviction check using the tenants SSN and drivers license (which you must have their permission to do) you can't be sure that the records you are viewing are in fact one and the same person who was your tenant. Easy results are not equal to accurate results.
The best relief is usually accomplished in person. Sit down and meet with the PM company and LISTEN to their side of the story.
I would say that being out $10k is probably on the low side for most full blown evictions where the unit was "trashed". While this is a significant sum, trashing a unit by damaging the paint and flooring can easily result in expenses over $10k plus the lost rent, court and attorney costs.
The only thing that you have said that makes me question the PM outright was the lack of a key for the sheriff to gain entry. The PM should have had a key to the unit, but again, that might not be the PM's fault if the tenant changed the locks against the provisions of the lease.
How is it that you consider the tenant to be "judgement proof"? It's not that hard to get a judgement. The challenging part usually is collecting on that judgement. After you get the judgement you have to garnish their wages to collect which means you have to follow them from job to job to collect. You can even get judgements against someone on disability or social security but you can't garnish that those to collect your funds.
Once you dig into it. It really boils down to time. How much time do you have to chase the money from the PM or from the tenant? How much is your time and mental energy worth to you?
If you have the time and desire. Got to the wall with the tenant. What you don't get from them, go after the PM for in the same manner.
Small claims is the best approach which has a limit of $7,500 in Colorado. If you sue for more than that, you have to hire an attorney and would likely end up paying the attorney more that you collected no matter who you sue.
With small claims it is set up for you to go mano a mano with the people that owe you money. You will have to appear in court (probably multiple times) and you said you are out of state. Think about the travel time and expenses. The people I know who are successful at the process are those that do it often and know and understand the system. Just learning the process takes a good bit of time and effort. Generally, you pursue the first one out of principle and apply the learning to subsequent cases. After about 3 or 4 you are at break even. After that, you know the process and can actually make it worth your while to pursue collectable tenants for the money they owe you.
The alternate option with the PM is to first threaten them with reporting to the Real Estate Commission and then if they don't play ball actually report them to the commission. I have been audited by the commission (the result was minimal issues to address) and it is no fun. It costs money to defend against the commission as I hired an attorney help me and it was worth it.
The final alternate approach to the PM company is also to threaten and again if no traction is made to contact the state attorney general about their business practices. The Colorado attorney general recently collected a $1M settlement with no admission of guilt from a large Boulder property management company. This was related to how the PM company treated tenants so you might not get as much traction with the AG's office since you are not a tenant.
Finally, you are obviously hurt and angry and want to be made whole. I can easily construct a scenario where the PM did absolutely nothing wrong despite your current perception. My advise is to get as close to seeing the PMs side of the situation as you can before you take any action against them. Of the PMs in the world, 10% are great, 80% are adequate, and 10% are bad and deserve to be spanked and put out of business. In my mind, nothing you have said, definitively puts them in the bottom 10%. It's possible, but it's also possible that is not the case. Consider the downside (what if you take them to court and end up with nothing) before you jump in the deep end.
I hope this situation resolves itself for you and doesn't put you off of investing in real estate. This is a bump. There will be others as well. No matter the track you take, note your lessons learned and apply them to the next situation and move forward.
You asked what I would do in your shoes. Meet with the PM and see it from there perspective. If they did not come of as 100% criminal, I would move on and mark up as a learning experience.