Actually, for a brief span of seven years there was such an army (or, strictly speaking, a corps) - the Palmach, founded in 1941. Its officers wore no special insignia (and in fact, there were no ranks in the usual sense, only command-titles such as "platoon commander"), got the same pay as the privates - and ate together with them.
This all makes a lot of sense if one recalls that the Palmach, while an always obedient non-political arm of the Haganah and the Yishuv was also a sort of self-conscious brotherhood, steeped in a rather radical socialist ethos. Indeed, in a famous headcount in 1948 Ben-Gurion (not quite an impartial observer, but still the figure seems to be valid) estimated that 60 out of the 84 top officers of the Palmach were members of the radical-left Mapam party. (For a somewhat different, culturological, take on this point see here).
Towards the end of 1948, Ben-Gurion, striving to establish the nascent Israel as a properly constituted state (he called it ממלכתיות, which can be very roughly translated as "stateness"), forcefully integrated the Palmach into the IDF.
Notably, the Palmach veterans were very incensed about the introduction of British-style officer messes, rank insignia, and separate pay grades for officers (one such letter is cited on pp. 176-7 in the book בראשית by Michal Tzur). However, their egalitarian system was probably incompatible with being part of an organized state's army and had to go, whatever were its merits.
Even so, this source indicates that joint officer-men meals persisted in ex-Palmach units for a while:
We remained in Beer Yaakov in August and September training and
preparing to go back to action in the Negev. During the months of
August and September there was a United Nations ceasefire. This gave
the Israeli Army time to re-equip and train for the next fight. The
Palmach was integrated into the Army, even though the officers still
ate with the soldiers in the same mess hall and would not wear any