Yes, TE amps have a 'sound'. But you can choose to change it. That's what the famed Graphic EQ is for. I don't use the pre-shape preset and I don't use the 'smiling face' EQ curve. Why would I? Whether or not they sound 'old fashioned' is up to the user, IMHO.
Does an Ampeg B15 sound old fashioned? Does an SVT sound old fashioned?
And yes, they can be quite heavy. A Trace combo generally weighs more than a modern combo. But it's not like one is immovable and the other is a feather. It's all relative. A modern combo might be 15kg. It's light - for a bass amp. But you still wouldn't want to walk any great distance carrying it.
In short, hauling gear is always a bit of a pain in the àrse at the best of times, whether your combo weighs 15kg or 25kg. For me, both would require wheeled assistance if covering more than around 50 yards, so the weight advantage is moot.
I'm not a Trace fanboi by any means, but the combo I have does exactly what I want it to do, the build quality is astounding, it's stood the test of time (20+ years) and it was a bargain!