Your approach seems to be the best you can do, since the models you've listed don't support any sort of integration with Alexa (and I think it would be infeasible to modify the TV/cinema system directly to connect it to your network).
The Logitech Harmony Hub seems to take a very similar approach of simply sending infrared signals to control 'dumb' devices (just like a traditional universal remote would), so it seems likely that your approach of using an infrared transmitter is correct (otherwise Logitech would have likely gone with a different solution to the problem!).
From the Logitech website, here's a description of how the Harmony Hub operates:
CLOSED CABINET CONTROL
Control your devices behind closed cabinets and doors with IR, Wi-Fi and Bluetooth® wireless.
For your specific use case, infrared (IR) is the only transmission type you need, since your TV and home cinema don't support the other methods.
However, on this note, it may be worth considering whether it's actually worth building it yourself as opposed to just buying the pre-built solution from Logitech for this purpose. I've checked compatibility for you using the compatibility checker and both your devices are listed as supported, and your method of using the Pi may be similarly expensive (but will also require the trouble of designing the software and finding the IR codes):
From the guide you linked, here are the other parts required:
(full model names omitted, and prices quoted as cheapest when this post was written for 1 unit)
When you total these, the price comes to approximately £57.70, which saves £41.30 compared to the Logitech Harmony Hub, but you will have to spend time designing the software to control the IR transmitter/receiver and the Alexa skill. Alternatively, if you decide to integrate it with SmartThings with the ZigBee/Z-Wave components you listed, the totals will be approximately £84 and £114 for the whole device (the latter is more than the £99 Logitech Hub!).
You may be able to reduce the price significantly, though, by using a more simple microcontroller, since I doubt it will be so complex that a Raspberry Pi 3 would be required.
In summary, if you're in for a challenge and don't mind getting your hands dirty, you could save quite a lot of money by designing it yourself (as long as you avoid the really expensive components). If you want convenience, Logitech's solution seems ideal since the integration with Alexa is already there and you also get control via your smartphone for free!