If a government anointed any given handful of OS organizations as preferred benefactors of donations, I'd expect grifters to infiltrate those organizations and parasitically siphon off the funds one way or another.
Incentives matter. Government incentives are to be popular, or attract the support of other people who are popular or influential. Being efficient or effective is only a small part of that. I don't know that there's a good solution to the incentive problem.
I’d expect companies like Raytheon, Cerner, Lockheed Martin, Boeing, and HPE/CSC/DXC to win a supermajority of those contracts.
Please check https://joinup.ec.europa.eu/collection/eu-fossa-2/news/how-c...
If the EU wants an open-source conferencing solution they have to do it in-house (whether from scratch or fork an existing solution) and treat it like a business with a clear objective and actual employees (instead of benevolent devs donating their time & effort) including positions which open source projects often deem unnecessary like UI & UX design, and so on.
I think France is also funding/developing Matrix.
There's also EU Public License (EUPL). One notable example of software that uses it is Pi-hole.
One more interesting thing I can think of is Joinup, whose idea is to share solutions between administrations in the EU: https://joinup.ec.europa.eu/