Since any new variant would be competing with Omic

Since any new variant would be competing with Omicron it would have to have a higher or equal transmission and/or reproduction rate and in your scenario also be deadlier. The higher transmission rate is almost certainly due to the higher reproduction rate. If you have a higher reproduction rate and a deadlier virus people will sicken sooner or die and will be taken out of the environment where they can transmit the disease.

Whether Omicron will cause more deaths because of the wider spread even if less deadly will all depend on exactly how deadly it is.