Often the Butterflies get taken off because somebody wants to renew the O-ring. Then when reinstalled they have only replaced the O-ring with no regard to the special circumstances about this part. Those small screws were originally staked and they should never fall out unless a former owner has messed with them and did not properly reinstall them.
Your repair is fine for getting it running. I think you should now rebuild both carburetor butterflys. To do this you will need new shafts (2), screws (4) and O-rings (2). If one of these screws gets sucked into the engine it can and most likely will ruin your day.
You should be able to reuse the round disk.
Check the budget and see what these all cost. When you are ready to do this we will cover the proper replacement of these screws for you.