Hi Klemmy, yes you're right that 3" rear tyre doesn't look too out of place on the ETZ250 at all. I think the 3" H-03 I have will not look silly given what yours looks like.
But I really didn't like the way the 3" Mitas H-03 felt on the front of my ETZ250 when I tried it
Either way, it doesn't actually look like you have all the much tread left anyway so I suppose it is up to you as to when to change. Believe me that I am not a big believer in tyre year codes but only for self interest. They are handy for giving the age of a tyre but I feel that fully exploring the tyre and looking at it every now and then and inspecting is much better than simply going from a manufacture year, potentially wasting lots and lots of money- I mean, I own 8 vehicles, imagine the cost of replacement tyres every 5-7 years? Of course it is the front tyres which last a long time though and they are the concern as time goes on.
Are the tyres still soft and maluable? I like to put my finger on a flat part of tread and see how much 45 degree pressure I apply in a rotational way before I feel it slide- some tyres I have picked up I think I will get rid of again because they are very hard and not much pressure is needed to make my finger slip. But the Mitas is in my opinion a pretty soft and forgiving tyre to age.
I actually have a 16/17 year old front tyre on my ETZ250 right now - infact I specifically bought it because I really like the tyre but it's not made anymore (or I can't get it locally) Michelin M45. But it's in great condition, flexible, crack free, feels great around corners, and also a 28 year old tyre on the front of my Suzuki TS185 - but I think I will be changing that soon more for offroading tread reasons plus maybe 28 years is actually pushing it more than most would accept. However this is just me- you accept your own risk towards using old tyres.
That said, when you look for new tyres sometimes you can get better deals with a 3" instead of buying a 3.5" or whatever. Sometimes the price difference really is quite a bit! Also I have encountered quite a difference in price on two tyres made by the same manufacterer but have different measurement, but the outcome in the same tyre. For example for me the Avon Roadrider II AM26 100/90x19 vs 3.25x19. The 3.25 is cheaper, and yet the size is exactly the same- seriously I have two one of each measurement side by side in my shed! So yes, look around! Also, I don't have any problem with using Asian tyres from Vietnam or Malaya. They're pretty good these days for the price! And actually the big brands don't necessarily last very long despite their name. Heidenau are much prefered by people who ride MZ's because they are a successor if you will to Pneumant - But I don't actually think much on these tyres, I don't much like their Pneaument copy designs but also they last a lot less time then competitors. The
half price cheaper Mitas tyres (for example) last
at least another 1/3 longer, at least for me!
That bike linked sounds perfectly normal to me... Best thing is to video your own vehicle and we can have a listen, but so far each concern you have had and mentioned the same sound in other bikes you have linked a video of - they all sound normal...