How can I automatically syntax check a powershell script file?

PS Script Analyzer is a good place to start at static analysis of your code.

PSScriptAnalyzer provides script analysis and checks for potential code defects in the scripts by applying a group of built-in or customized rules on the scripts being analyzed.

It also integrates with Visual Studio Code.

There are a number of strategies for mocking PowerShell as part of unit tests, and also have a look at Pester.

The Scripting Guy's Unit Testing PowerShell Code With Pester
PowerShellMagazine's Get Started With Pester (PowerShell unit testing framework)

You could run your code through the Parser and observe if it raises any errors:

# Empty collection for errors
$Errors = @()

# Define input script
$inputScript = 'Do-Something -Param 1,2,3,'

[void][System.Management.Automation.Language.Parser]::ParseInput($inputScript,[ref]$null,[ref]$Errors)

if($Errors.Count -gt 0){
    Write-Warning 'Errors found'
}

This could easily be turned into a simple function:

function Test-Syntax
{
    [CmdletBinding(DefaultParameterSetName='File')]
    param(
        [Parameter(Mandatory=$true, ParameterSetName='File', Position = 0)]
        [string]$Path, 

        [Parameter(Mandatory=$true, ParameterSetName='String', Position = 0)]
        [string]$Code
    )

    $Errors = @()
    if($PSCmdlet.ParameterSetName -eq 'String'){
        [void][System.Management.Automation.Language.Parser]::ParseInput($Code,[ref]$null,[ref]$Errors)
    } else {
        [void][System.Management.Automation.Language.Parser]::ParseFile($Path,[ref]$null,[ref]$Errors)
    }

    return [bool]($Errors.Count -lt 1)
}

Then use like:

if(Test-Syntax C:\path\to\script.ps1){
    Write-Host 'Script looks good!'
}

I stumbled onto Get-Command -syntax 'script.ps1' and found it concise and useful.

ETA from the comment below: This gives a detailed syntax error report, if any; otherwise it shows the calling syntax (parameter list) of the script.