I am aware of LSDJ and have tried to use it via various GameBoy emulators, but I find the interface to be incredibly clunky and am wondering if anyone has seen a suite of software that takes better advantage of the keyboard and/or mouse interfaces commonly available to us today, but that can generate the same sort of "chiptunes" sounds?
You should try LMMS, which has a SID and a FreeBoy plugin.
There are a whole bunch of ways to make chiptune music in modern environments.
The "classic" approach to this is to use tracker software to sequence chip samples. Most trackers use samples, and there's a lot of history to using chip samples in tracker music. There are a few traditional trackers available for modern OS's, such as MilkyTracker or Modplug Tracker/OpenMPT. There are also more modern trackers with newer features, such as Renoise. Trackers are neat because you work at a very low, fundamental level with your samples. It can feel like programming as much as composition.
A more common approach these days, since DAWs and plugins are so common now, is to use a plugin to either emulate the sound of the old chips, or play back samples of them. If you're on Windows, try Tweakbench's free Toad, Peach, and Triforce plugins which make sounds based on NES waveforms, likely similar to what you were making with LSDJ. This approach is great too, because you can easily combine chipsounds with other recorded or synthesized sounds, and it's generally very easy to add effects and make broader edits. There's also a larger community of people working with this kind of software that you can rely on.
Of course if you REALLY want to be authentic, it is possible to sequence notes on actual chip hardware, using a solution such as a HardSID or MSSIAH cartridge. There's no more authentic way to get these sounds, but recording them can be a hassle - a lot of those old machines were noisy, and it can be hard to get a clean signal.
Pretty much any "tracker" program (Impulse Tracker, CheeseTracker, etc.) can be used to make chiptunes, given fundamental waveforms as samples. There are also a lot of chip-style softsynths available for sequencers such as Cubase, Logic, Ardour, Reaper, etc.; Logic even comes with a whole bunch of them with different wonderful bleepy-blorpy characteristics.
On macOS, you can try GarageBand (free) or Logic Pro X ($200). You can import your own custom chiptune instruments and have at it from there
On Linux, macOS AND Windows is another Application called LMMS (free and open source), however the interface is extremely clunky from my experience and I personally would only consider it as a last resort.