Some things may be more trouble than they're worth. I have spent hours and days DIYing a problem that is a 1 hour job for a professional. You don't pay a professional for that 1 hour. You pay for their 20 years of experience.

If you've never extracted a screw before there's a certain amount of finesse that is needed especially with such a small screw. I used to be a heavy vehicle mechanic. Worn, weathered, stripped, and broken screws were an everyday part of the job. To be honest the first 10 times I had to do it I may have messed up every single time. With experience that little screw would be drilled out, tapped,and done in no time. But if you mess up you're paying for the repair of the whole valve anyway.