Confirming this issue. I experienced it with an SE 2020 and it is common to several Plantronics headsets in the series. M70, M50 and others. It connects correctly and then at exactly 15 seconds, the signal drops. Sometimes it reconnects automatically, sometimes it has to be pushed or forced to reconnect. It is an endless loop and the phone is useless while driving.
This was tested with multiple headsets, multiple models and then each headset was immediately confirmed to be working fine when used with an iPhone 6, an iPad Air2 and other Apple and non-Apple devices. iOS 12 (on the older comparison devices) and 13 were both tested with no differences noted.
The iPhone SE 2020 has several other issues that make it almost unusable at times.
There were so many flaws with the first handset that I assumed it was defective and Apple replaced it. Most of the same issues are present with the second handset as well. I have never received an inherently defective product from Apple over 25 years of loyalty, but the iPhone 2020 in its current state is not worth of the Apple brand. I am hoping that a software upgrade to perhaps 13.6 can address these issues (or 14). If not, this phone is almost a brick to me. If others are experiencing similar issues, I would hope for some major restorative action from Apple.
Other issues with the SE 2020 include a very jittery scroll process with the screen with the document or page in ANY application jumping back rapidly to the top on a hair trigger touch.
In addition, at least one mission critical accessory (genuine Apple) that works fine with other iPhones - the lightning to headphone & power 2 in 1 adaptor is not compatible with the 2020 and one apparently needs to purchase them all over again.
Having had the headphone jack taken away when it was crucial for various Pro Music accessories, was very difficult to endure. To not only have to purchase adapters, but to have preexisting adapters that work fine with iPhones 7 through 11 but not wi the SE is an extremely disrespectful treatment of the customer base.