I'm not so sure that 'having a constitution' is a problem in itself, so much as 'having a rubbish constitution', which, as you say, means 'one which was written in a past age and is unfit for purpose in the modern world'.
The UK system also finds itself hampered by anachronisms, though it does at least allow parliament the flexibility to adapt legislation in line with changing times.
I think the ideal would be to have a relatively modern constitution - as with all countries that have emerged from dictatorship or colonialism in the past 80 years or so, with an enforced revision process every 50 years, say, to weed out parts that have become obstacles to fulfilling the needs and wishes of contemporary citizens. That said, Chile, for example, has had a pretty patchy and divisive experience of revisiting its constitution over recent years.
Here in Spain, the 46-year-old text we have works fairly well, but is ready for a slight revamp.
That's the problem with constitutions - they become enshrined as sacred cows, and are then difficult to shift, hence the need to build in an upgrade process from the get-go. Though that demands a level of humility for which politicians, senior judges and academics tend not to be known.