The service account should not need to be a local admin, and I agree with you, I think it's a bad idea.

Here is the description of the SQLServerSQLAgentUser$MYCOMPUTERNAME$MSSQLSERVER group on my server.

That seems to point to the exact issue you're having, getting the service account the necessary permissions at the OS-level without granting Administrator.

I just ran a search on all ACLs on my machine and this is what my local group is able to access:

1. Full Control over C:\Program Files\Microsoft SQL Server\MSSQL10_50.MSSQLSERVER\MSSQL\JOBS

2. These effective permissions on C:\Program Files\Microsoft SQL Server\MSSQL10_50.MSSQLSERVER\MSSQL\Log

Traverse Folder

List folder / read data

Read attributes

Read extended attributes

Create files / write data

Create folders / append data

Write Atributes

Write Extended Attributes

Delete subfolders and files

Read permissions

You could create your own group, grant it these permissions, add your non-admin service account to it an give it a go. It may not get you there since there could be registry permissions or other local policies in play but it's worth a shot. If not then you could consider opening a ticket with Microsoft....or simply uninstalling and re-installing from scratch.