I'm a Senior Property Manager and I do some work for my clients.
However, I am well aware I do not have the skill set to do any major work and if it's something I could do at home by myself, than I am happy to do it for my client.
Also when tenants vacate and they haven't cleaned one wall, rather than calling them back and arguing with them about it, I just wipe it down myself.
I carry a broom, a mop and a cloth around with me. As well as a hammer and a small screw driver and Allen Key set. If it requires anything more than what I have, than I won't do it. If it involves any kind of risk, ie getting up on a ladder or on the roof or dealing with electricity etc, I won't touch it. A door handle, yes. A toilet roll holder, yes. Lifting a plug off a shower drain, sure no problem.
However, I've never had a client ASK me to do it, I've just done it and often not told them. Don't want to toot my own horn to the client
When at Routine Inspections, I will also do what I can. It shames the tenants in to having half a brain when they watch you quickly tighten a screw on a loose towel rail.
The difference here is though that I am building this business and whilst I have the support of another person funding the business and paying me a salary, a level of pride comes in to what I do. My clients treat me well and with respect and so in return, I am happy to go above and beyond. Any growth for our business and a good reputation benefits me greatly in the long term too.
I do worry about the insurance side. If your PM doesn't feel confident in doing it, I wouldn't push it. If they fell and the screwdriver went through their hand, that wouldn't be a nice position to be in for anybody. This type of stuff does happen and if you've pushed an issue that is outside of their job description, well, no fun at all really. The insurers may come looking for you.
A tenant may just fix it themselves which would be great for everybody. In the meantime, do you know somebody who would be happy to go and do it for you for a dinner voucher or something?
I would also worry about setting the precedent that there is an expectation that minor works will be completed by a PM.
Hope you get something sorted.