The Eurosport app only runs on Samsung TVs that are using version 3.0 or higher of the Tizen operating system. Given that v3.0 was only released in May of 2017, and that it would probably have been several months before TVs using it reached retailers, that means that the Eurosport app won't run on any Samsung TV that's more than about 3 years old. So how old is your telly?
You don't need to subscribe to Amazon Prime TV in order to access Eurosport Player. You simply need an Amazon Fire TV stick, enabling you to make a one-off purchase (and, of course, pay for your Eurosport subscription), rather than having to make ongoing payments to Amazon.
https://help.eurosport.com/hc/en-gb/articles/360009615660-What-are-the-devices-supported-by-Eurosport-and-Eurosport-Player-
The basic ('Lite') version of the Amazon Fire TV stick costs £29.99
https://www.amazon.co.uk/dp/B07ZZW7QCM?ref=MarsFS_SMP_shl