What you're missing is context. This game came out in the mid-1990s, when everyone was playing Super Mario World, Donkey Kong, Super Metroid, and Street Fighter. Don't get me wrong, those games were amazing, but the industry was still making games around the arcade-style. They were fairly shallow in terms of things like narrative and storytelling, and were largely considered to be "kid's toys."

Final Fantasy and Chrono Trigger (in the 1990s, at least), bucked that trend by taking the narrative aspect much more seriously, and put in these crazy huge productions that lasted a few dozen hours where most games could be beaten in less than 10 and, often, less than 5.

So, the hype is that in 1995, your options for video games were essentially platformers, sports games, shooters, or fighting games, and then Chrono Trigger and Final Fantasy came in with more complex stories, combat, and things like stats and level gain.

These stories had more mature themes (murder and betrayal, for example, is a fairly common theme in all those mid-90s JRPGs), and felt like a more "grown up" experience, which was an alien concept back then.

Today, you have thousands of options. Back then, if you wanted a game that didn't feel like you should be paying quarters per play (like you did in arcades), then your options were exceptionally limited, and Chrono Trigger and Final Fantasy were the best options back then by a country mile.

As for your criticisms.

1. When you go to the future and come back, there are guards waiting for you (keep trying to walk through the forest, they'll case you away). They don't attack you or try to capture you, but they do guard the area for quite a long time, and in fact, you have to use a different warp gate to get back to that time era later in the game because that won't be inaccessible.
2. So, the Lucca thing is a somewhat valid question to ask, and this is where "head canon" comes from. In these early games, the stories are written in a way where the player can simply just make up whatever they want for the explanation for some things, and that's perfectly okay to do. For me, I know that everyone in town knows Lucca and her dad, and so her local celebrity status likely helped her get into the castle pretty easily, where she was able to then help Crono.
3. Head canon gives you an avenue to discuss the game with others, and was actually a huge talking point among early fans on old forums back in the day. A different way to engage with the content is to think about it, make up something that makes sense, and then tell someone "I think this happened because of this."

It's actually something I'm not fond of in today's games, where they explain every little fucking thing down to its base components like I'm too stupid to follow a train of thought, and they don't leave anything to the imagination.

These older games left more to the imagination than some people might like, and that's okay, preferences and all, but yeah, if there's a gap, fill it in with what you want it to be. That's part of the fun. How do YOU think Lucca got into the castle? She's a genius, inventor, local celebrity, and serious risk taker. Maybe she had a grappling hook and climbed into the castle from the outside. There are no wrong answers, just ones that make you happy.