Are SQL Server's temporal tables here to stay? We're considering different approaches to save records' history in an application.
Some of the managers in charge are concerned that as this is a relatively new feature of SQL server, and Microsoft may cease to support it.
Are these concerns justified? 
Is there a roadmap to this feature that says what Microsoft intends to do with this feature in the future?
I don't think the concern is legitimate. This feature is introduced in SQL Server 2016 and it's very clearly mentioned that it applies to SQL Server 2016 and later versions. Later version meaning for now it is at least SQL Server 2019 which has been released.
Temporal tables

Applies to: SQL Server 2016 (13.x) and later, Azure SQL Database, Azure
SQL Managed Instance  - Azure Synapse Analytics
SQL Server 2016 introduced support for temporal tables (also known as
system-versioned temporal tables) as a database feature that brings
built-in support for providing information about data stored in the
table at any point in time rather than only the data that is correct
at the current moment in time. Temporal is a database feature that was
introduced in ANSI SQL 2011.

If at all, there is chance of removal of any feature or anything they do mention at Microsoft documentation, for instance &quot;Database Mirroring&quot;. You can read the details about the same as below:

Note
This feature is in maintenance mode and may be removed in a future
version of Microsoft SQL Server. Avoid using this feature in new
development work, and plan to modify applications that currently use
this feature. Use Always On availability groups instead.

I hope this addresses your concern.
Note: Given temporal tables are part of the ISO/ANSI SQL standard, and it's highly unlikely the feature will be removed from SQL Server.