This is The Genetic Fallacy - you’re confusing what they were, their origins, with what they have become, what they are now.
If the Labour Party is moving to the space formerly occupied by The Conservatives, then it’s a good idea for them to disassociate themselves from strikes. They will be the party of paternalistic capitalism.
Absolutely, another fallacy is to a) conflate the part of the poitical spectrum occupied by the big three parties with the spectrum itself and b) that these parties occupy the middle or centrist ground of that spectrum.
Labour have demonstrated, in the breach, what their relationship with the unions is on innumerable occasions. Their cynicism is matched only by that of many of the unions themselves. They deserve each other while the average working person deserves neither.