Lead crystal as the name suggests has a thin lead coating on the glass, today it will not be lead, but it will still have a coating, the dish washer tablets have chemicals in them that can remove the coating so gives them a whitish tinge, you can use additives to replace the coating.
Glass is not all the same, the coefficient of linear expansion varies, with heat proof glass it is very low, so since it expands so little it will not be shock damaged, however since a dish washer is cold fill and heats the water up slowly they will not get shock damage, glass is never fully solid, Roman windows have been shown to be thicker at bottom where the glass has slowly flowed down the pane, but it is extremely slow.
Heat proof glass will really mess up recycling as it will not mix with the rest, this is why broken glass is not accepted for recycling although rather pointless as it gets broken on the wagon so little chance of sorters getting rid of it.
The glaze on pottery is like a glass coating, so the dish washing tablets can also affect them, however the glaze crazes anyway from the stress of pouring in boiling water, and so all the dish washer really does is remove dirt from the cracks so you can't see it.
So with pottery may as well wash in dish washer, it does make them cleaner, with glass it is better to hand wash, it stops the coating coming off, but it is not the dish washer that causes the coating to come off, but the tablets we use in the dish washer, so a new dishwasher will not help, it may help adding glass washing additives or changing the tablet.