Transportation from La Spezia to Monterosso that's not the local train

Personally I don't think train would be a nightmare but nightmare has a different meaning for different people. There is lots of trains and it could be crowded at that time of year. I would not avoid it but your travel style could be different.

I haven't been to the Cinque Terre for about 35 years, so I'm not going to speculate about the conditions on the train between La Spezia and Monterosso. I would like to point out, however, that the entire trip takes between 18 and 23 minutes, depending on which train you catch. Many urban commuters spend far more time than that in packed subway cars, ten times a week. I would not worry about this unless one of your travel party is mobility-impaired or physically fragile.

If you're willing to wait until 10:35, there's an Intercity train that goes directly to Monterosso with no stops in the other Cinque Terre towns. Cost is 8.50 euro per person and it would avoid the crowding issues you're concerned about since seats are reserved. You could wait in the train station caffe for an hour and enjoy a nice cappucino.

Edit: You could buy tickets for the 10:35 Intercity train a day or so ahead if you wanted.

The regional trains start from there at La Spezia Centrale, so you'd have some possibility of boarding early and getting a seat. Monterosso is not the end of the line, but many regional trains (such as the 9:25 and 9:55) are only continuing one more stop to Levanto, so I would expect less crowd getting off.

It's not a mosh-pit. Even at peak times, your fellow passengers are civilized, and since the Cinque Terre Express in 2016 was begun the trains have been much less a concern. Also at 9:11, you should be ahead of cruise ship passengers if there's a landing that day. La Spezia Radio Taxi lists the taxi tariff as 80 Euro.

Think you are worrying over nothing. Be on the platform early and get in quickly if there are lots of people on the platform. Many people will get off at the stops before Monterosso as in general people want to be in the smaller, more scenic towns and Monterosso is less scenic and much flatter. So by the time you get there, a lot of passengers will have already gotten off. Lots of people have luggage on this line. Didn’t find it an issue at all. And it is a short ride. Try to get seats on the coast side for more scenic views.

Is taking a cab or private transfer practical?

Rail transport is about 20 minutes

Rubber tire transport close to an hour ( google's estimate---which in my experience is usually optimistic)
Is this 'practical' to you?

I was there today. We took our first train from La Spezia at 9:25 and it was a zoo! Totally packed. But I assume later would be even worse. We originally were going to go to Monterosso and work our way back but decided to get off at the first town (mainly because we were in such a rush to get on the train I forgot to validate our tickets so we wanted to get off ASAP and validate them). By mid day almost every train was delayed and it was hot!. IMHO if you can get the direct train to Monterosso, do it. You are arriving at a good time and crowds will be at a minimum compared to later in the day.