I have followed the P365 since its introduction and may very possibly buy one at some point in time.
While it is pretty clear that the majority of P365 owners have not experienced significant problems, it is also clear that a substantial percentage of P365 owners experienced one or more teething issues, some of which were catastrophic in terms of rendering the pistol immediately non-functional without prior warning. The most commonly discussed have been the broken strikers and the dislodged trigger bar springs, and some of these issues occurred after very substantial round counts. But there were others including sights that fell off, dead night sights, barrel peening, failures to return to battery, and extraction issues.
Some of these problems were addressed by SIG when they temporarily shut down P365 production to redesign the recoil spring assembly and by switching from the original Siglite sights to X Ray sights. Then there was the redesign of the striker, and some modification in design or assembly of the trigger bar spring. I am aware of 4 different polls of P365 owners conducted on various forums, some of which were quite detailed. One can talk about the unreliability of internet polls and selection bias and such, but the results of these were strikingly similar in that between 20 and 25% of P365 owners had experienced a problem severe enough that they had to return their pistols to SIG for service at the time the poll was conducted. Even if there was considerable selection bias in these polls, I would still consider a critical failure rate of even 5% to be unacceptably high in a carry pistol.
But things do seem to be settling down now, in no small part to the owners of early P365 pistols who willingly or not became SIG's beta testers. The most common issues I have heard reported in the last 3-4 months have been failures of slide lock on empty magazines, and a number of light primer strikes with the redesigned striker. The slide lock issue could, of course, be largely due to shooter error, but some experienced shooters have had this occur when they were quite certain they were not riding the slide lock lever. The light strikes might be of greater potential concern, but these issues seem to be much less common than the earlier critical failures that thankfully seem to have abated.
I have shot the P365 and agree it is very shootable and accurate considering its small size, and it has an impressive magazine capacity. I will probably give SIG another 6 months or so before I spring for one, but I do suspect that one is probably pretty safe buying current production P365.