I don't think I'm old (late 40s), and I recently spent £26 on what I'm sad to see is the last printed GBTT from Middleton Press -- I will very much miss having it in future. Of course if travelling at a weekend I will always check online for engineering works (being able to do this is a big improvement over pre-Internet days), but this might involve just looking at NRE to see if there are any works that might affect my journey, and if not, then looking up times in the printed timetable. I don't remember being badly caught out as a result since online information about engineering works has been available (except one occasion when the work still wasn't mentioned on NRE just a few days before).
There are two separate discussions to be had -- printed vs. electronic, and matrix timetable vs. journey planner. A journey planner can be useful for some purposes, but it is not any kind of substitute for a proper timetable -- the former will tell you what the computer thinks you should do to travel between two particular points in a particular time range on a particular date, but if you want to get a general idea of what services are available so you know what journeys are feasible on future occasions, check in case you want to come back at a different time or on a different day, investigate alternative places to change (which may be pleasanter places to wait or give you more chance of making the connection), investigate alternative start/end points (e.g. 'is it worth getting the bus to somewhere with a more frequent train service?'), see if you can save any time by getting an 'unofficial' connection, see when the next train is if you miss an official connection, and so on, it gets tedious making multiple searches when a proper timetable will show you everything at a glance. It's like how a satnav, useful though it may be, is not a substitute for a map (or a knowledge of geography).
The paper vs. print question is less clear-cut, as a matrix timetable is essentially the same in both formats and it's a question of which you prefer or is more convenient. I'm presumably going to have to get used to finding my way round the online pdfs of the GBTT and printing those I use most often -- I never got used to it during the brief absence of the printed version a few years ago, but I might eventually. As for individual route timetables, yes you can print them off, but a properly produced leaflet is more convenient than some home-printed pages, and although I've not much gone in for printing railway timetables, my experience of bus timetables is that they aren't always designed to be printed conveniently and legibly at home.
It seems to me that it's actually new or occasional users who have most to gain from a proper timetable if they can get past the barrier of not being used to it (in the same way that it's people with little knowledge of geography who are most at risk of driving to the wrong end of the country if they put the wrong postcode into their satnav). Having used the printed GBTT for nearly 40 years I know where the railway lines go and have a general idea of what kind of service is available on most of them, but in the interval between Thomas Cook announcing the end of their European timetable and the timetable staff reviving it, I wondered how I would be able to plan travel in countries whose networks I was less familiar with.