A high-speed connection between Square One and Toronto Downtown would be very well used. Unfortunately the Galt sub / Milton GO will not be employed for that purpose, as it is the CP mainline and that traffic will not be redirected in the foreseeable future.
What are the other options?
1a. The Line 2 West extension, as the thread title says. Straightforward, but not really high-speed, riders going from Square One to Toronto Downtown will have to endure the Line 2 milk run with stops every 700 m, and then transfer at St George.
1b. A variation of that: Line 2 extension at first, but built in such a way that the segments can be disconnected at Kipling or at Cloverdale. Later, a new express subway line (stops every 2-4 km) added between Cloverdale and the downtown, and the Square One segment gets connected to that new line, while Line 2 terminates at Cloverdale. This is more expensive than 1a, but will add a large amount of new capacity into Toronto Downtown, and serve both Mississauga and South Etobicoke.
2. A tunneled branch off the Lakeshore West GO line. Sounds crazy at first, but actually will only require a 5-km long tunnel from Square One to the southern corner of the Lakeview Golf Course (between Dixie and Ogden). That's shorter than the Line 2 West extension, and the trip between Square One and Toronto Downtown using that connection should take no more than 30 min. The drawbacks are the space taken from the Lakeshore West corridor, and having few stops within Mississauga other than Square One.
3. An extension of Ontario Line westwards, eventually reaching Square One. That extension can have more stops and provide better local connections than a GO branch, but more stops will mean a longer travel time. And, there are concerns regarding the Ontario Line capacity. The Exhibition terminus of OL should play an important role in intersepting the GO riders and relieving the Union station. But if OL gets extended and the trains arrive to the Exhibition station being full, then few GO riders will want to transfer there.