Loire Valley without a car

Use AutoEurope for your car searches. I would rent a car.

I would rent a car for greater freedom in seeing all there is to see in the Loire Valley area. You won't be tied to bus schedules, etc. However, there are better options for car rental in Tours (St. Pierre de Corps station). It is only about 21 kilometers from that station to Amboise. And there are TGVs to Tours from Paris that take just over an hour. (I am assuming you will be going through Paris on your train ride from Beaune.) Driving from chateaux to chateaux is easy in the area, and you have the option of seeing other towns also (e.g. Saumur) and the troglodyte caves.
Whether a car is cheaper than bike rentals and public transportation in the valley, I don't know, but a car is, IMO, definitely preferable.

P.S. I would use Autoeurope for your car rental search. They will show several rental car companies. In our experience they have been comparable to renting direct from the individual company and they are well-run and organized and have good English-speaking customer service back-up.

Rent a car! I cannot imagine doing the chateaus with out one

I agree with renting an auto! Time is money after all and you don't want to take forever waiting for connections or getting run over on your E-bike! Also autos can carry your luggage and your picnic basket. Trains are great for getting somewhere long distances faster, but autos work best for stopping at charming places on the way. Flix buses are OK for getting from bigger cities to other cities. If you are used to driving at home; I would always reccommend driving yourself for the best experience.

Allow me to pile on and strongly recommend that you rent a car, both for convenience, and ease of visiting exactly the places you want to see, on your own time. But I wouldn't rent in Amboise, but at the St Pierre des Corps station in Tours.

Firstly, you will have more rental agency options located just outside the station. Secondly, if coming from Beaune, the route with the connection in Nevers goes first to St Pierre des Corps before connecting to Amboise. If you take the route to Amboise that connects in Paris, you have to travel to a different station to make that connection, which is annoying with luggage.

But before booking, take a good look at the agency's open hours. Most close Sat afternoon and Sunday, and during lunch.

One more vote for renting a car. As noted by previous posters, you get to be on your own schedule, bring along whatever extras you like (picnic baskets, for example), and add in spontaneous stops as you please. One more thing. In any town, the train station is not necessarily located right next to the point of interest that made you want to go there. You might be faced with figuring out how to get from the train station to the chateau, cathedral or whatever - and then back to the train station. I think a car is definitely the way to go.