Constructor was not found when adding a TilesViewWebPart with Powershell

This would have worked with other types of webparts, e.g.

$PLwebpart = New-Object Microsoft.SharePoint.WebPartPages.XsltListViewWebPart

However, the TilesViewWebPart class is abstract, which means that you cannot instantiate it.

You would have to do something like creating a class library to implement the webpart, and then use the .dll to add an instance of the class you created.

Class

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace TilesWP
{
    public class TilesWP : Microsoft.SharePoint.WebPartPages.TilesViewWebPart
    {
        protected override Microsoft.SharePoint.WebPartPages.TileData[] GetTiles()
        {
            throw new NotImplementedException();
        }
        protected override string ViewTitle
        {
            get { throw new NotImplementedException(); }
        }
    }
}

PowerShell

Add-Type -Path TilesWP.dll
$webpart = new-object TilesWP.TilesWP

You'll probably find it interesting to see an actual implementation of this.

Note:

If you actually don't want to create a custom implementation of the tiles webpart, but all you are trying to do is display a promoted links view, you can use XsltListViewWebPart

$webpart = New-Object Microsoft.SharePoint.WebPartPages.XsltListViewWebPart
$webpart.ListId = $tilesList.ID
$webpart.ViewGuid = ($tilesList.DefaultView.ID).ToString("B").ToUpper()
$webpart.ChromeType = "None"
$webpartmanager = $web.GetLimitedWebPartManager("/Pages/Home.aspx",[System.Web.UI.WebControls.WebParts.PersonalizationScope]::Shared)
$webpartmanager.AddWebPart($webpart, "Header", 1)
$webpartmanager.SaveChanges($webpart)