user123, for what it’s worth, I flew with an Italian ID card and had no problems leaving the UK with Easyjet or British Airways. I did this five times (as I was without my passport for 13 weeks): to France, Spain, Belgium, and Italy (twice). I am fairly certain EU law requires the airlines to accept ID cards for travel to the EU, and that your partner would be eligible for EU-261 compensation if she were wrongly denied boarding.
Your partner’s bigger problem could be getting back to the UK if she does not have Settled Status. If she does have Settled Status, follow the steps I listed above. Carry digital and physical copies of the following EU guidance and show it (if needed) to the border officials for the exit check and the airline:
https://ec.europa.eu/info/sites/default/files/faq_travel_with_id_cards_final_for_publication.pdf
I had no problems whatsoever entering or leaving Italy, always explaining that I had settled status / permanent residence on exit. No problems with Belgium, though the immigration officers asked if I had a passport so they wouldn’t have to key in the ID card data. No problem entering Spain; on the exit, I had to go through a long explanation. France was a mess: at first, they didn’t want to let me ENTER, saying I needed a passport to travel from the UK (ignoring that I was presenting an EU ID). Then I was initially rejected on the exit check and by Easyjet, and had to get supervisors both times.
The UK border was easy. The officials are trained on the ID card / Settled Status process.