Personally, I care about it, but don't lose any sleep over it.
I do the regular things many home/craft brewers do. Here is my process. It works pretty well for me. Others do it differently with similar results, which is great. For me it has become habit, so I don't really think about it much.
First, I try to get my water ph right to get all the flavor and sugars out of my grain. In my area I need to add calcium carbonate and lactic acid to the mash/sparge. This may not help with clarity, but if for some reason it doesn't turn out clear, the taste will make up for it. I add Irish moss towards the end of the boil, whirlpool 10-15 minutes after the boil, then chill fast with a counter-flow chiller.
I leave it in the primary fermenter for a minimum of at least 8-10 days (even if the SG is stable). Then I cold crash for a couple days in the fridge (or more if I have time) before bottling or kegging. I haven't used a secondary in a while. It settles well enough in one vessel. The only time I have added gelatin during crashing was when I forget the Irish moss.
I am careful when I move the beer to the keg or bottling bucket so as not to disturb the trub. I usually ferment in buckets with spouts which are above the yeast cake/trub so this isn't really a problem until the last quart or two of beer. I try to move the bucket out of the fridge as carefully as possible to not disturb the trub/yeast cake while moving. I then let it sit for a little while to settle before moving on to the next steps. I usually do 8 gallon batches split between the keg and bottles. I transfer into the keg first and the remainder into a bottling bucket. When I am at the bottom of the fermenter I watch closely when I pour the beer into the bottling bucket and stop when right at the trub cake. When I am bottling I add priming sugar in the bottling bucket during transfer, cover the bottling bucket and let it sit for a few minutes or half hour to let any large particles that may have transferred settle and the sugar dissipate. I know that the first pint or two in the keg might have a haze (not usually the case) but the rest should be clear if it isn't a wheat beer.
I like it when my beers are clear when they are supposed to be. It is a point of pride. I would bet that most (all) brewers hold their filled glass up to a light to see if they can see through it. If they say they don't they are lying.
When I bottle condition, unless it has been in the bottle for a while so the yeast cakes up solid, inevitably some is going to stir up during the pour or just from being moved around. I realize there isn't much in an individual bottle, but it is going to be there. I pour carefully if I think about it, but if not, I RDWHAHB. The people I give my beer to usually are aware of this. If not, they don't care after the first taste. Bottling from the Keg will help with this if you are giving it away, but I tend to bottle and keg at the same time so don't usually do this.