DBA Permissions on LOCAL SERVER containing SQL SERVER
There are a number of US government regulations that, depending on how your company interprets them, require this separation of duties.  Sarbanes–Oxley (SOX) is the big one for publicly traded companies.  HIPAA  for healthcare organizations, etc.

The first question I have is, of those tasks you outlined, how many are you doing directly from the server?  Arguably many of those tasks don't require local server access, but I understand this would be frustrating if you are used to performing this work locally on the server.

Regarding workarounds, my suggestion is to be an adopter of this policy.  Do this by making sure that your security team is properly locking down user accounts, but creates clearly defined policies for service accounts and that those accounts are setup appropriately.  I would even encourage you to look at Group Managed Service Accounts as this makes everyone's life easier if you can make them work in your environment.  Follow this up with moving whatever processes you can off of the servers.  There are real security concerns with users actively logging into servers regularly, and I would say you should avoid doing this unless absolutely necessary.  Argenis Fernandez, during one of the most entertaining PASS sessions I've attended, even showed that anyone with access to a local admin account on a server can get elevated access to the SQL Server regardless how "locked down" it is.  Honestly, you could even go so far as to request all of the servers get configured to run as Windows Server Core, which is about as secure a Windows server as you can have. Finally, document those processes that cannot be moved off or need to have exceptions created for them.  This will likely be your biggest headache in the short term, but proper documentation should allow your organization to bend the rules to fit the business needs.

Look at it this way.  If anything goes bad, such as a breach of information, and it's traced back to unauthorized access to a server, this policy may save your behind without you knowing.  I've worked for years in the healthcare and insurance industries and as much as a headache as this sort of policy may seem to be, it's really a good policy to adopt.

This process will be frustrating... at least it has been the number of times I walked a similar path, but don't make it a we vs. them issue, instead approach road blocks with the following statement "this needs to happen so the business can run, how do we accomplish that within these restrictions?"  With this approach, it's not about you or your job, it's about your company staying in business which makes it a we vs. the policy issue.