you mentioned that the games have to run on both desktop and mobile devices, but how do they decide if a game can run on mobile devices? There a lot of mobile devices with different hardware specs. Some devices will run a game smoothly and others will lag and might not even run the game at all.
Usually publishers have a QA department and will validate the game before finally paying the developer. They will likely test your game on the latest (or close) generation iPhone, iPad and Google Nexus.
Of course there is no way you as a single developer can verify your game will work without any issues on every available device. Even a bigger company would struggle to achieve this 100% I imagine.
Also I believe you can ignore performance problems with outdated devices. Noone will expect your HTML5 game to run great on some older Android Gingerbread smart device.
Remember to support different aspect ratios and resolutions with your game. It's one of the most crucial things for mobile development. In many cases this can be achieved with the proper fullscreen option in Construct, if your game is properly designed at the same time.
Still you obviously should have access to at least one Android or iOS device, ideally running the latest available version. Test your game there and you should be good.
I use a Google Nexus 7 (1st generation) for my mobile testing. Had a few concerns from the QA before, but it's mostly stuff you can easily adjust. I admit it can get tricky (in my case) if the problem is solely iOS related, since I do not have access to such a device.