There are many guides and discussions on alternating vs. single-tapping, with advice like "alternating makes high bpm jumps easier, but single-tapping makes accuracy easier on slower jumps", but here I just want to compare and discuss the pros and cons of full- and semi-alternating. Would like to hear other people's experiences and thoughts on it. : )
TL;DR: My preliminary conclusion (which may change upon hearing your arguments/experiences) is that an expert in full-alternating should be able to play fast complex tapping patterns with more ease than an expert in semi-alternating, but that it's much harder to learn to be an expert in full-alternating. For some people, the benefits of full-alternating may not be worth the extra time it takes to be good at it. I'm not confident in this conclusion at all, and results could vary for different people.
By the way, if you want to learn full-alternating, McOsu has an experimental mod that forces you to do full-alternating (McOsu can be played without VR). There's also this program that uses an AHK script to force full-alternating in regular osu! (you won't get banned even when using it ranked). Also, this and this are good maps for practicing alternating (playable on half-time).
Terminology
Single-tapping is when you use only one key (e.g. X) for everything except streams.
Full-alternating is when you alternate between X and Y for everything. So if you look at the key counter on the right side of your screen while playing, the K1 and K2 counter will always be the same or differ by 1. (If the counter isn't there, you can reactivate it in the osu! settings.)
Semi-alternating is when you alternate between X and Y except for some complex note patterns. For example, suppose there are three consecutive triples in the song. Full-alternators would tap them with XYX YXY XYX, whereas single-tappers and semi-alternators would tap them with XYX XYX XYX.
Arguments for semi-alternating
• Semi-alternating requires you to learn fewer base finger patterns, so it is easier to learn and may be less complex to play (or maybe not, see discussion below). Here "base finger patterns" are things like using XYX for triples and XY or YXfor doubles. Consider the example from earlier, where you have to play three consecutive triples. The full-alternator uses two different base finger patterns (XYX and YXY) whereas the semi-alternator only has to use a single base finger pattern three times (XYX). The semi-alternator can choose to start every stream, quadruple, triple and double with X, whereas the full-alternator has to learn and use twice the number of patterns--both the patterns that start with X and the patterns that start with Y.
• Almost no top players use full-alternating. (Rustbell and Azer semi-alternate.) And since the top players tend to know what's best (goes the argument), full-alternating is probably not best. The counter-argument is that this argument is a base rate fallacy, and the reason most top players are not full-alternators is simply because it is very hard to learn and few people try it out. So even if full-alternating gave a significant advantage, we would still expect few top players to be full-alternators. The other counter-argument is: Bubbleman and Vettel. They're full-alternators and they're GOOD.
• Full-alternating is really hard to learn. I personally started out with single-tapping, and then after half a year, I decided to learn full-alternating. I found it difficult, but I eventually learned to do it instinctively (not saying I was good at it, I'm just saying that it felt like my main playstyle at the time). But I eventually drifted (unconsciously) towards being a semi-alternator because it was slightly easier and my brain reinforced the behaviours that were slightly better at producing dopamine (i.e. hitting patterns successfully). So even if you feel like full-alternating is your main playstyle, it could take longer to consolidate it as your permanent playstyle.
Arguments for full-alternating
• It feels smoother. Semi-alternating feels more like stuttering through complex tapping patterns tbh. Full-alternating feels smoother than semi-alternating, and that may mean that I need to practice semi-alternating until it feels just as smooth, or it may mean that full-alternating will always feel smoother even for experts at both styles. And I'm assuming that how smooth the different playstyles feel affects performance on complex tapping patterns (i.e. that it is important for the skill known as finger control). This is actually my main intuitive argument in support of full-alternating, but it's not a very strong argument.
• Full-alternating fast complex tapping patterns requires less complex hand movements, and may therefore be easier to play fast once you know how (or maybe not, see discussion below). Consider the three consecutive triples again. A semi-alternator would play them XYX XYX XYX, but this requires more hand movement than XYX YXY XYX because the semi-alternator would have to lift their X-finger before every triple whereas the full-alternator is perfectly set up for doing an YXY right after doing a XYX.
• It's cool. Almost no top player full-alternates? That just makes players like Bubbleman and Vettel cooler. They are unique flowers amidst swathes of grey blooms.
Which style enables higher speed?
It seems to me like there are two ways your speed could be limited. Either you have physical limits (i.e. your tapping-hand can't tap faster, or your aim-hand can't move fast enough) or you have cognitive limits (i.e. you're too slow in figuring out which instructions to send to your hand, or you can't read the patterns on the screen fast enough).
I argued earlier that full-alternating strains your hand less, because it requires fewer movements through complex tapping patterns, so you should be able to play higher BPM with full-alternating without approaching your physical limit than with semi-alternating. But I also argued that semi-alternating requires you to send fewer types of tapping-instructions to your hand (e.g. XYX XYX XYX is only one type of instruction sent three times), so maybe it enables you to play higher bpm without approaching your cognitive limit? Maybe, but I think the truth is a bit more complicated than that.
I think that, yes, in the learning phase, full-alternators will be limited by their cognitive speed limit for longer. They will have more types of tapping-patterns that they need to practice, so it will take longer to become good at them. But I also think that once a full-alternator and a semi-alternator has reached expert level at their respective tapping-patterns, the full-alternator will have the advantage. Why? Because the full-alternator will need to use fewer hand-movements (as explained earlier) and that means that they need to send fewer muscle-activating instructions from the brain. So if the cognitive speed limit is determined by the size and number of the muscle-activating instructions, then full-alternating enables higher speeds.
(For the unnecessarily technical-minded: the Kolmogorov complexity of the instructions sent by a full-alternator may be higher for any given osu! beatmap, but the speed-limit of the brain doesn't depend on theoretical Kolmogorov complexity, it only depends on the number of synapses and the total axonal and dendritic length it needs its action potential to traverse, and the length of that pathway which is myelinated and... ಠ_ಠ)
So I conclude with a graph I just made up. (Key assumptions: Full-alternating takes longer to master, but is superior once mastered. The difference between the two playstyles is small once mastered. The benefits of each unit of practice decreases over time.)