Running a search engine is capital-intensive in the sense that to have a search engine, you have to maintain an index: a copy of all the text of the publically viewable internet, updated regularly. This requires a lot of network bandwidth and a whole lot of disk space.
If you tried to run an index on GCE it would cost you 5x or 10x what it would to do on servers you own. If you managed to create a viable product with this incredibly onerous constraint, to the point of posing an actual business threat to Google, then they would just shut you down, since all your stuff is on their servers.
Generally, cloud stuff is more expensive than running on bare metal: https://blog.pinboard.in/2012/01/the_five_stages_of_hosting/
If you tried to run an index on GCE it would cost you 5x or 10x what it would to do on servers you own. If you managed to create a viable product with this incredibly onerous constraint, to the point of posing an actual business threat to Google, then they would just shut you down, since all your stuff is on their servers.