I want to eat = Θέλω να φάω. The construction να+verb is the subjunctive (https://blogs.transparent.com/greek/the-subjunctive-mood-in-modern-greek/). Notice how the verb endings match the subject of each verb. For another person: Θέλω να φάει = I want him/her to eat. The way the verb is conjugated in the subjunctive is a bit like saying I want that he eats, I hope that he will have returned, ..that they arrived earlier etc. It's not great grammatically in English with 'that' but it gives you an idea of why the verb needs to change depending on its subject as we're not simply using an infinitive form.
Exactly! I want to eat = Θέλω να φάω. I want να I eat
N.B. Don't think that να (na) means to. There is no direct translation of να into English. I think it is better to think of να as a word that has a certain function, one that in English is expressed with the infinitive. Να also has other functions, which you may encounter as you learn more Greek.
Here is an example showing that 'να' (GR) is not the translation of 'to' (EN): I can eat = μπορώ να φάω. There is no 'to' in the English sentence; it would even sound strange if we were to add it.
I don't agree that it is "more or less" an infinitive. The infinitive of a verb is by definition an adverbial noun, similar in meaning (and somewhat in usage) to the English gerund. In contrast, the to + verb construction used in Modern English and in Mod Greek (and the other members of the Balkan Sprachbund) is its own beast with its own rules. I am not sure if this holds in Mod Greek, but in Mod Bulgarian and Macedonian there is a neo-infinitive construction which is distinct from the to + verb "pseudo-infinitive": e.g. 'Имам ядене да готвя' (lit. "I have eating to prepare" or in normal English "I am about to prepare something for eating") with ядене (eating) being a "neo-infinitive" vs 'Имам да ям сготвеното' (lit. "I have to eat the prepared [food]") with да ям (to eat, present 1p sg) being what you call a "more or less" infinitive (Note: this is the same as the Greek να + verb). For this reason, it's not advisable to mix the two concepts...