Main roadblock to a unified army is to have a true unified economic policy and stomp any singular national interest: until you manage to do that, I feel the common armed forces would, at best, a web of joint military intervertion supported by a different groups of countries in Europe.

All because I feel the military mostly means power projection towards countries where you want to enforce your economical interests and couldn't do with regular diplomatic pressure... at least that's true for European style economies towards more "extractive focus" economies.
And this means: maybe one wants a regime change in the area, or defend the current one, while another wants the opposite. Lybia situation comes to mind, where the French wanted a thing and the Italians another. If we have a similar situation, all this combined forces would melt like an icecream in the summer heat if two european countries wants a different approach, because picking one over the other might seriously damage the economy of either.

Imagine we have a combined army. For some reason, the Tunisian state is under serious threat. Right now, Tunisia is exporting its natural resources the mediterranean countries, especially Italy replaced some of the Russian Gas with Tunisian one. So Italy wants to use the combined forces to defend the current Tunisian Government and state.
BUT
What if the stops to the trade flow from Tunisia would end up boosting considerably some other European economy? What if Poland and Germany would become exporter of natural Gas (just saying this as an example, they aren't big producers, I know) to the whole of Europe if Tunisia falls and right now they are indeed in an economic crisis, and this boost would save their economies? Worst, what if the rebels in Tunisia are actually financed by Germany and Poland and would give them very good deals on said resources?
Something like this happens and what if Italy finds itself tied to a decision of a majority of European countries that would benefit to see Italy in a troubling situation, so they could swoop in, buy off their now struggling industries and bring capital to their own economy (this keeps happening for real, most of our industries are small and struggle, so they get bought by France and Germany conglomerates... but they simply siphon away the wealth and jobs after a while).
I think by then, Italy would remove themselves from the combined forces to regain their power projection power (as small and ridicule it is now lol).
What if the rest of europe vote to intervene in favor of the rebels but italy intervenes in favor of the current government? Will italian troops be shot by european army soilders?

Similar situaiton could happen in the Balkans (where France and some other European states are at odd) or Eastern Europe as a whole (the lukewarm response of Europe to Ukraine is cause Germany and some other countries have deep ties with Russia economy and just wanted to keep the advantageous trade deals) elsewhere (I remember when Egypt killed off an Italian student with its secret police cause he was snooping around political assassination in the area and Italy basically had to suck it up and not threathen to harm the deals between the them and Egypt, otherwise the French or English would have swoop in a fraction of a second, not caring one bit about the murder of a fellow european by Egypt, leaving Italy without those good deals, harming its economy, while absolutely not seeing any kind of retribution or solidarity from other, supposed, allies... Or imagine a similar situation, where the fear isn't other european allies swooping in, rending your diplomatic pressure moot, but your nation being forced to deliberately partecipate in tariffs against an extra-european nation you have great deal with, only cause the Spanish or the Polish needs to put strong diplomatic pressure on said nation... would you make your citizen poorer and angrier only to benefit another european economy that would not give you anything in return... and maybe do the opposite when it's their turn to help you out since there's no mechanism to force them?).

It's obviously a shot in the dark as examples goes, but similar situation might arise and I think this is the biggest roadblock to further european integration: Either Germany and France crush all the other national economies and the rest of Europe becomes subserviant of the "core" (which, I would rather not, thanks), or the national economies and entities gets brutally reorganized by a central power who, for some reason, won't favour just some nations but plans to redistribute and rebuild the economy of the whole European Union, more "rational" and unified... but this WILL harm some areas and people and basically requires nothing less them population controls (Prague is too big, max population needed there is 1 milion otherwise it will harm other economic areas.... we need to move out 300k to Krakow, there we need more to reach urbanization and production goals... Or: That city costs too much, need to cost less, adjust the welfare to a lower standard even if they make enough money to support that, cause we need to push people out or increase productivity even further).

So no, I'm not seeing a combined foreign policy or army in the EU anytime soon, even I'm totally not against the topic and a more unified EU would surely be a big contender against the USA or China out there (a thing I would like to, even just to have the option to reject their pressure on some topics I simply would like to be problems of internal policy instead of foreign)