Deleting multiple channels at once

This is a task best done with scripting.

The following "simplistic" script will remove all alpha channels, which may or may not always be a good idea (for example, you may have spot channels that you wish to retain while bulk removing all other alphas):

if (documents.length > 0)
{
docRef = activeDocument;

for (var i = docRef.channels.length-1 ; i >=0 ; i--) {
if (docRef.channels[i].kind != ChannelType.COMPONENT)	{
$.writeln(docRef.channels[i].kind);
docRef.channels[i].remove(); }
}

}
else
{
alert("There must be at least one open document to run this script!");
}

The following script will remove all standard alpha channels while retaining spot channels (code taken from a larger script, see below):

if (documents.length > 0)
{
removeAllAlphaChannelsNotSpot();
function removeAllAlphaChannelsNotSpot() {
    // https://community.adobe.com/t5/photoshop/remove-alpha-channel-during-a-batch/m-p/11285697?page=1#M347558
    var myChannel = app.activeDocument.channels;
    for (n = myChannel.length - 1; n >= 0; n--) {
        var value = myChannel[n];
        aC = myChannel[n];
        if (aC.kind == "ChannelType.MASKEDAREA") {
            aC.remove()
        }
    }
}

}
else
{
alert("There must be at least one open document to run this script!");
}

https://prepression.blogspot.com/2017/11/downloading-and-installing-adobe-scripts.html

Look for the 30th July 2020 update at the end of the topic thread:

https://community.adobe.com/t5/photoshop/free-script-remove-selected/m-p/10104624

After holding down the Shift key as you clicked on each channel to add to the list you chose to delete, did you then click on the Trash symbol? You would then get the notice "Delete selected channels?" Choose Yes.