Welcome to the forum!
May I suggest the following thread on
Basic Concepts and Techniques as a starting point for refreshing your basic sailing knowledge. Feel free to ask specific questions on issues of how to sail the boat in your own thread in the "Seamanship and Boat Handling" section of the forum or the thread "Everybody is an expert sailor?" which tries to collect the seemingly "dumb" questions that people may feel to shy to ask.
The mast is not very heavy, about 25lbs or so, which means that supporting it at the back in a mast crutch like you describe is quite reasonable. Mine is about as minimal as you can bet, it's simply a bit of 1/4 with a notch at the top. I do strap down the mast in the rear. First, I have a PU strap (bungee would work as well) that holds the mast to the crutch. Then I tie a very long bungee from one of the stern cleats, once around the mast, and to the other stern cleat. That cuts down on the bouncing, but I like to also tie a bit of parachute cord from cleat to cleat and around the mast. That one, because of the inevitable slack that develops, would allow some bouncing, hence the bungee, but it acts as the fail-safe in that it positively limits how far the mast can go.
You correctly noticed that most powerboats are secured to their trailers with two straps from the transom. That method is very secure, but can't be directly replicated for the DS. On my trailer, there are attachment points for straps right in front of the tail lights. I hook a ratchet strap into them that I lead over the boat, about 4' forward of the transom. That will keep the boat from flying off the bunks at the rear of the trailer, but, because it's a single strap, it does allow the boat to shift a bit. I trailer routinely up to about 50 miles and it's not been an issue. I do stop en route and check trailer bearings for temperature, and the boat for shifting. One note on the strap: make sure it has some twists in the free span across the cockpit: a fried of mine, a former trucker, advised me that this will cut on fluttering in the air stream compared to a nicely "flat" strap.
I also tie a strap over the bow, making sure to hook it under the aft edge of the foredeck cleat. Without that, the strap can slide forward over the bow: while it doesn't look it, the bow narrows just enough to make up for the longer distance, so if your boat does not have a cleat in the middle of the foredeck you would have to install one before using a strap across the bow. This second strap acts as a fail-safe on the winch strap in holding the bow to the trailer. (Because I don't take the painter off the bow eye, I also wrap it around the winch post and cleat it off on the foredeck. (That still leaves the bow eye as a single point of failure, hence the second ratchet strap).
The forward end of the mast is supported by a wooden support I added to my trailer winch post. Some trailers come with integral mast support. I use the tail end of the painter to tie the mast to the foredeck cleat (with spiral wrap leading forward that would grip and hold the mast in the event of a very sudden stop) and also have a bungee cord to the mast support to eliminate bouncing. Again, I wouldn't trust the bungee alone to positively hold the mast.
My boat has a third support for the mast in the middle of the deck. Many people do not have that, but I like it because it limits swinging as the mast bends int the middle. I tie the mast to the cuddy deck with a line from one of the cleats on the aft end of the cuddy to another (not all DS models may have those cleats). There you have it. I tend to worry more about trailer bearings overheating or failing than about the boat or mast coming off and over a moderately long tenure with the boat, I've indeed have had issues with the former and never the latter.
In this instance, I don't think that you were overthinking things, but you correctly noticed a rather unsafe setup. With a single point attachment, if the bow eye fails (and one of mine failed spectacularly on my second or third outing, right when I wanted to pull the boat out of the water
), the boat easily separates from the trailer and being heavy with a smooth bottom will travel considerable distances on its own before it comes to a stop.
Being new to boat and trailer, I advise you to repack your trailer bearings (unless that was positively done by the prior owner right before the sale), and if the hubs aren't equipped with them to install "bearing buddies" - those are spring-loaded hubs that hold a reservoir of bearing grease under slight pressure, making water ingress while launching the boat that much less likely. They original brand may no longer be in business, but you'll find equivalents from other vendors. They are held in by press fit so you need to measure your actual (not nominal) hub diameter with great precision with a set of calipers. (One of mine is just a bit out of tolerance, but will hang on if I use the green thread locker). I did not know about this and at the end of my first season was rewarded with a spectacular show of smoking rubber when one of the bearings seized up !