Do different speaker stands can really make a big sound improvement?

Not sure if Mitch2 was being sarcastic, but he had a definite point when he mentioned that a real audiophile would suspend the speakers from the ceiling with springs. Well, I was a real audiophile when in college in the mid 1970’s and had a girlfriend who was quite the macrame artist. She wove for me harnesses for my EPI 100 bookshelf speakers made with eight ropes to hang them from the ceiling.  With 2 ropes per corner the speakers were securely nested so much that I could tilt the speaker back and forth for proper time alignment of the woofer and the tweeter.  The improvement in openness and soundstaging was enormous by the standards of the day.  For about the price of a 12 pack. FYI the source was a Thorens TD124 with Empire 980 arm I bought for $20 from a radio station converting to tape cartridges, and an Ortofon VMS 20E cartridge.  Amps were all tube, a Dynaco PAS 3X preamp and the matching Stereo 70 as a power amp.  I really miss that system…

It happens but not always so as is usually the case: it depends.

The devil is always in the details when it comes to application of science and technology. Generalized deductions can easily get you in trouble.

When I bought my ProAc Response 2's 25 years ago, they came with matching custom-made MDF stands that looked wonderful but sounded awful.  I finally found a used pair of Sound Anchors, and it wasn't until then that I was able to stop fiddling with the speakers and really enjoy them.  There's no question in my mind that speaker stands make an enormous difference.  Personally I would take the manufacturer's recommendations seriously if at all possible in terms of space and available funds.  It would strike me as very odd if a company as old and reputable as Harbeth were promoting "snake oil" solutions.  They have every interest in making sure their customers hear their products at their best.

My Townshend podia make a huge difference. Not quite your question but those podia can have a stand set upon them. For example:

Yes they do.

Check to see if Resonant Woods of New Hampshire is back in business (they were without a wood shop last time I checked which was 1 year ago). They are similar and high quality but a bit cheaper than TT. 

Proper joinery / quality wood work is expensive and like or not, buying the appropriate stands for your Harbeth's is the price of entry.

Comparisons with other speaker brands / stands is irrelevant.