I will echo your story about P series motherboard replacement. I had two separate onsite repair incidents to attempt to fix a hard lockup/freeze issue on P1. I won't go as far as to say the onsite support is useless, but the process seems flawed, in terms of logistics and the materials.
After first motherboard replacement, as soon as service tech left, the laptop would no longer power back on. Plugging in the A/C adapter, the power light would blink 3 times, but the machine would never POST. After trying a series of tricks with reseating/resetting CMOS battery on the mobo, I gave up and contacted the onsite technician. He had left my house less than 2 hours from the repair when I reached him. He told me he wouldn't/couldn't come back out until a ~separate~ MTSA ticket had been filed for the new issue (now laptop POST). The fact that I had to re-raise an MTSA ticket was annyoing. After contacting cust support via phone, they entered a new MTSA ticket. A few days later, tech came back onsite, and replaced the motherboard the second time. This at least enabled the laptop to POST again, but the freeze problem wasn't corrected.
I will say the onsite tech was friendly (even though he wouldn't come back out right after the first failed motherboard replafement, which I gather he was just following procedure). He did quickly do the replacements, and I got the sense he was knowledgable and not the root cause.
What I think the main problem is what the OP mentioned: the repair techs are being sent out with refurbished replacement parks that are 50/50% chance of being utter junk.
Somewhere an accounting decision is being made that the reputational damage of risking customer alienation from lengthy service disruptions does not outweigh the cost savings of using refurbished parts for onsite repairs. I would urge Lenovo to rethink that policy. IMO every onsite repair should be using fresh, new out-of-the-box components.