Last year during the summer, I would go home on weekends.
As OP correctly says, past habitual - he usually/always went home on weekends. No problem.
Last year during the summer, I would have gone home on weekends.
As TimLymington says, native speakers might be tempted to interpret this as an incomplete statement, with the unstated continuation "...but because of [some excuse] I didn't". But that's only because we're struggling to find any way to make sense of a rather unlikely utterance, so I suggest we consider a simpler one...
Last Friday night I would have gone to the pub.
Again, it's possible there's an unstated "...but my wife wouldn't let me."
But if the speaker had just been asked "Where were you last Friday night?", it would be a perfectly normal and complete reply. In such a context, the speaker may be implying that he's normally in the pub on Friday nights, so he would have been there on that particular occasion. Or he may be implying that although he's not absolutely certain, it would turn out that he was there, if the matter were to be investigated more thoroughly.
Last year during the summer, I would have completed three projects every month.
This one somewhat complicates the issue, because three projects every month automatically creates a past habitual context. Native speakers don't feel tempted to assume there's an unspoken "...if I hadn't lazed around the pool every day.", though they might expect a follow-on along the lines of "but I think I could only manage two projects a month this year."
The fact of the matter is that whatever the textbooks say, people often use would have [verbed] in that way. Semantically, there's no real difference between OP's version and...
Last year during the summer, I would complete three projects every month.
...or indeed...
Last year during the summer, I completed three projects every month.
As pointed out, these are not good examples for getting to grips with usage of would [have], because phrases like during the summer and three projects every month blur the focus on exactly what time we're talking about, and what exactly might have been habitually repeated.
A bigger problem with all forms of 2 is that the second comma shouldn't be there.
In "However, they didn't know that Brian had flown to Brazil the week before, and had been abroad when the burglary took place." the comma breaks up the list of things "they" didn't know, which was one intended purpose of the "and". The "and" now makes the "had been abroad..." refer to the "they" instead of "Brian".
The "was abroad when the robbery took place" is a direct consequence of Brian's having gone to Brazil, and is the other purpose of the "and" (i.e. "and therefore").
A sentence shouldn't start with "however", either, so a better construction would be:
"They didn't know, however, that Brian had flown to Brazil the week before and had therefore been abroad when the robbery took place."
Best Answer
Perhaps by some. If so, it's the sort of opinion that can work perfectly well in guiding ones own style, but would make someone a fool if they started prescribing it as one everyone should follow.
Really, no common form is "weak" in and of itself, and developing a superstition that you should avoid some will in fact produce weaker writing because you'll be avoiding what would serve you best and using something less appropriate instead. Having personal preferences when forms are close to each other though, is reasonable, probably unavoidable, and part of what makes for ones personal style.
The simple past is generally terser, which can make it punchier, and so make it stronger in a lot of cases, but that's not always the case.
It doesn't (there are other possible additions that could make it clear, but on its own, it doesn't). In this regard it is less clear than the habitual past. Which would make it weaker, would it not?
Personally, I'm inclined to favour a combination of the two:
This has some of the terseness of the simple past, but also the more specific framing-in-time of the habitual past. With no further context than the two sentences, this seems to me stronger than either alternative (though not so much so that I'd consider someone crazy for disagreeing).
In a wider context, I might change my mind again.
(All this applies to the used to habitual past, too).