The carrot approach has worked well but uptake amongst those in 20s is significantly lower.
Their is a sensible plan to do this, where you never force vaccination but you encourage it with the stick.
I do think the government should mandate all public sector workers get vaccinated or find alternative employment. Similarly they can encourage companies and organisations (including universities) to take similar policies but never force vaccination. Now their is obviously a time issue with many students unlikely to get fully vaccinated by October, to combat this they can simply require a negative lateral flow test done same day to enter universities for semester 1. (and hopefully a big enough inconvenience that they will get jabbed).