stored procedures – Connection server timeout – Not queried

This is just a guess, but I would try to enable the delayed schema check for the linked server under heavy load:

EXEC master.dbo.sp_serveroption @server=N'Linked_Server_name', @optname=N'lazy schema validation', @optvalue=N'true'

If this option is set to false, SQL Server checks the default value
for schema changes that have occurred since compilation in Remote
Tables. This check is done before the query runs. If anything changes
In the schema, SQL Server recompiles the query with the new schema.

When this option is set to true, schema validation is performed on remote tables
delayed until execution. This can cause a distributed query to fail
with an error if the schema of a remote table has changed between
Query creation and execution.

source

To check only the connection server's metadata when you query the linked server itself.

Another workaround could be to use dynamic SQL statements to query the linked server

DECLARE @SQL nvarchar(max)
IF (1=1)
BEGIN
SET @SQL = 
N'SELECT * FROM OPENQUERY(LinkedServer,''SELECT * FROM master.sys.databases'')'
EXEC SP_EXECUTESQL @SQL
END