I have worked with old wiring like that. Quite honestly I turned away jobs that had wiring like this because I found that no matter how carefully you handle the old wiring the old insulation just falls off into your fingers. It just crumbles and you are left with bare wires. Just changing out a light fixture like you show can cause an electrician to spend so much more time fixing/repairing the existing wiring so the install is safe than it takes to actually change the fixture with more modern wiring. The problem is if I work on something like this and the insulation starts to fall off the wire then it is up to me to repair it and that is a job of all jobs sometimes because no matter how careful you are at handling the old wiring the insulation just keeps coming off further and further down the wire. Just moving the wires sometimes causes the insulation to just fall off. So I learned to just avoid it when ever possible even if it means passing on a large job. Unless it was to replace the old wiring with new I would not dare to touch it. Same thing when it came to working on a Federal Pacific StabLok panel. If I went to a customer's house and found this type of panel I just passed on the job. My fingers were not going to be the last fingers in that panel before it failed. The only way I would touch a FP StabLok was to replace it all together.
Yes, you can get another opinion from another electrician but I don't think you will get much more of a different answer.
The electrical system in your home is very important. You say it is a new house. New to you, meaning it is a very old house. I am surprised that your home inspector did not say anything to you about this even in general sense. Almost all home inspectors will not take apart fixtures, receptacle, switches etc when doing their inspections. They will normally advise you in general as to how old the wiring is and whether it should be replaced. They also concentrate on the electrical panel.
This is a decision you will have to make. I had to do the same thing with my roof. If I did not put $20,000 into it then I would never have a dry house and would have mold non stop in the house.
If your credit is fairly good check with your bank that you have accounts in. Find out about a Home Equity Loan like I did for my roof. The interest rates are far lower than personal loans or credit cards and much easier to handle each month.
Yes, with older wiring like that the chances of something happening is far greater than with new wiring. Only you can know where your comfort zone is by living with it or replacing it.
Take a picture of your electrical panel and post it here. Depending on what we see we may encourage you to go ahead and rewire including the panel. I know it may be something you don't want to hear but safety is important especially if you have a family and pets living there not to mention yourself. I would certainly becareful what you say to your home owners insurance company about it. They may force you to rewire or drop you.