So, users are locked into MePIN's proprietary app and depend on MePIN's website to log in, rather than using a open standard that can run offline. If you use HOTP/TOTP, you can use open source Google Authenticator, DuoSecurity, libpam, or any number of clients.
In MePIN's defense, DuoSecurity also has their own push notification for a single-tap login, so users are willing to trade interoperability for convenience.
Of course user behavior has to be considered. The MePIN app does allow the user to set up a personal PIN code, so an authorization would then require the PIN code and a tap.
If a user is willing to press the button, a PIN isn't going to stop them. Your app is decreasing security in favor of usability, which is not something look for when they are looking to implement two factor auth.
I think anyone who would blindly use your proprietary two factor solution that makes it easier for end users to authorize other people to log in would be silly.