The math works out, but you have to be honest with yourself about whether you'll actually use the machine. €2.50 a day is around €900 a year - so a Delonghi really does pay for itself relatively quickly. That said: good espresso beans from the supermarket cost maybe €5-8 per kilogram, and you only need 18-20 grams per cup, so a single cup really comes out cheap. That's the financial part and it checks out.
The real problem is maintenance and reality. Espresso machines need regular backflushing, the portafilter has to be dry, limescale buildup can become a nightmare - and honestly, a lot of people forget about this or don't do it carefully enough. The machine needs to be looked after after every use, otherwise the espresso eventually tastes like nothing or doesn't run properly anymore. That takes discipline, and not everyone has it. If that puts you off, then going to the Italian place is less stressful and you just pay your €2.50.
But if you're the type who regularly cleans their equipment and doesn't ignore the manual, then it's definitely worth it. Only then you shouldn't go for the cheapest option - better to invest in something a bit better, because the budget machines will frustrate you faster. Maybe test it out at a friend's place or a café first to see if the effort won't be too much for you.