The Electronic Times recently reported that Google wants to invest ~$880 million in LG Display for future production of OLED displays. If this rumor is true, I suspect the potential strategic investment would not just be for securing displays for future Pixel devices, but for helping LG Display to seriously re-enter the mobile OLED market. The company previously sold flexible OLEDs to LG Electronics for its G Flex and G Flex 2 smartphones in 2013 and 2015, respectively, but I am not aware of any other smartphones ever using LGD OLED.
To date Samsung Display has been far ahead of everyone else in mobile OLED due to its vastly greater investment in the segment. LG Display could probably make reasonable OLED displays, though, if it had the financial incentive to make major investments in smaller panels. It has already proven its OLED capabilities with its Apple Watch displays, however difficult they were to make, and of course its leading W-OLED panels for the TV market.
This rumored change in strategy would probably be more about industrial design demands than display quality considerations. Several vendors, including Google, want to be able to compete with Samsung and Apple’s (upcoming) bleeding-edge smartphones that strongly associate OLED displays with high-end industrial design. While OLED is not necessary at all for creating a design with minimal bezels, some or even most of these vendors likely require OLED because they want to bring curved displays to market, sacrificing some image quality in the process.
To be clear, there are many advantages (and some disadvantages) to working with OLED displays over traditional LCDs from an industrial design point of view, which I won't fully enumerate here. One of the major differences, while it sounds obvious, is that OLEDs do not have LCMs (liquid crystal modules).
No matter what, it's not at all clear that investing in OLED over LCD long term would be a smart move, and most display suppliers remain skeptical of the former. OLED is better overall now, but it has strong downsides in terms of lifetime, costs (due to lower yields), and various quality deficiencies such as severe off-angle color shifting and chromatic aliasing. LCD meanwhile constitutes the lion's share of the market. microLED won't come to market for years, but it has greater potential than OLED should its production become economically feasible.
If vendors bothered to pay for high quality displays, we would see smartphones other than those from Samsung with correctly calibrated OLEDs with leading-edge quality. Perhaps a vendor or two other than Apple may one day do that. For now, given Samsung Display’s massive lead I remain skeptical that anyone can compete with it on quality over the next few years.