I've owned several single axle caravans before switching to a t/a model.
I've never had any problem here in the UK with sites 'not liking' my t/a and caravan forums carry very few posts about problems in Europe although on some continental sites pitch size can be an issue.
Pros:
More capacity, full sized front benches to stretch out on, fixed double bed and a full sized rear washroom with lots of room to move around in and for storage.
A properly loaded well maintained caravan whether t/a or s/a should be inherently stable so no discernible lack of stability or gain in stability for me.
The unit is heavier so is less likely to be seriously affected by other overtaking motorway traffic.
The caravan is slower to respond to steering inputs when reversing because of the distance between the tow hitch and the axles so reversing is easier.
I use levelling ramps and have no problems with side to side levelling but I haven't got Alko wheel clamps.
Cons:
The most obvious is that an extra axle means more weight which can impact on fuel economy.
A t/a is not something that one could manoeuvre by hand so either reversing skills have to be spot on or a motor mover would be needed, which brings an added weight penalty.
If the t/a has Alko wheel lock receivers fitted there can be difficulties in lining up the wheel-clamps, jacking up the caravan is usually necessary.
A t/a caravan is usually longer so the back end will swing out more and this has to be considered when manoeuvring.
Storage, particularly at home can be a problem unless the property has a large driveway, and caravan insurance premiums can sometimes be higher because of a perceived increased risk of theft.