EU imposed rules mean that public sector bodies can't bid for rail franchises.
No.
The only stipulation is that infrastructure must be accounted for separately from operations, and that operations are periodically competed for (which we interpret as franchising).
Cross-border services are regulated by the EU (as opposed to national governments), much as air and ferry services are.
A level playing field must be established for new entrants, which means transparent access charges and a licensing regime (ie no closed shop).
Even SNCF/DB etc have only a limited licence to operate trains, and are both heading for a regional model.
Germany has public operators from one region running trains in another (not DB).
The rules were not "imposed", we voted for them (and to some degree invented them, as we were first into open access and franchising).
They are part of the single market legislation we were so keen on until yesterday.
There is nothing whatever in EU legislation to stop public bodies operating trains.
But they have to operate commercially and competitively.
Even SNCF and DB operate commercially, even if with massive government support, and are facing competition on all sides.
The reason we don't allow public sector bids was originally so that the entire industry would be out of public hands.
The expertise now sits with the franchise owners.
So there aren't any public sector operators, bar DRS (which is a creature of the nationalised nuclear industry, but is still a commercial TOC), and TfL which has no remit outside London.
To extrapolate further, if we stayed in the EU, and Labour "nationalised" its first franchise, it would still have to be competed in the open market.
Franchises can of course be changed and merged, to form larger operations, but they still would have to be commercial, not secret arms of the state like BR was.