The Crosstown LRT will make it much easier for people to avoid the Yonge line though. They'll only be a short trip from the Spadina line. If Yonge-Eglinton is too busy, I am sure many will skip it and transfer at Eglinton West instead.
It may even be quicker to take the Spadina branch, if your destination is in western downtown. Currently continuing on Eglinton across Yonge on a bus is pretty difficult because you have to change buses at Yonge-Eg station, and often buses stay in the bus bay for like 5-10 min as the driver takes a break. With the LRT/subway that trip will be much faster since it's underground, continues through the station, doesn't need to spend time going in and out of the bus bay through a very congested intersection and stops far less than the bus.
According to the Eg Crosstown website, it should take 6 min (4 stops) to go from Eg-Yonge station to Eg-West station.
It also expands our network to be more distributed and less dependant on a few subway lines. For example, if there's some sort of incident at say Davisville shutting down the subway from Eglinton to Bloor, then people from North York on Yonge could head down to Eglinton, switch to the Crosstown and take it west to Eg West, then take the Spadina line south to Bloor. As we build out the rapid transit system it should distribute demand more evenly and create more redundancy.
It also makes trips that don't go downtown much easier. Ex. someone from Eglinton in Scarborough could take a fast trip west to Eg West, then transfer to the Spadina subway north to York U. Or someone at Yonge & Eg could take it west to Mt Dennis, transfer to UPX to get to the airport.