I always had the idea that you can strum a bass guitar. An idea I got from watching Mark Hoppus play. Recently the bass teacher I know told me that you mostly play lead lines with a bass. I'm not sure what to believe.
The bass intro from the song 'carousel' always gave me the idea you could strum a bass.
Pretty much every string instrument can be strummed. The Blink 182 example you provided shows an electric bass guitar being strummed. Here is an example where Charlie Haden strums on double bass:
(the strumming part starts at 3:26)
1) Lemmy of the band Motorhead strums power chords all the time.
Here is Lemmy demonstrating his technique
2) There is a more elaborate technique for strumming on bass guitar which involves several fingers of the right hand, rather than a pick. The technique is called rasgueado, which is a technique taken from flamenco guitar.
The technique has been utilized by master jazz bass guitarists including Victor Wooten and Anthony Jackson. Chris Wolstenholme of Muse uses a similar technique in “Panic Station,” visible here:
This is not a standalone answer, Todd & Shev have covered strumming - but as for lead lines, they too are possible.
The idea that a bassist must simply play along to what the rest of the band are doing I think is neatly put aside by this band.
Before Royal Blood came along, if anyone had told me that you could make a 'proper' rock band with just a drummer & bassist, I'd have been mildly amused but unlikely to take them seriously.
If you go to the YouTube page, there's a track list with clickable links to listen to bits of each track, for a quick precis of what they do.
If it has more than one string it can be strummed. Keith Emerson has even strummed the harp inside his piano. I have to say that full strumming a five string may sound a bit funky with that low B, but rock bassists have been strumming for over 40 years.
And as far as "Playing lead lines is not usually in a bassist's remit. Neither is usually playing runs.", all I can say is Tim has obviously never listened to Jack Bruce, Tim Bogert, Busta Cherry Jones, or the host of other rock and jazz bassists that made the bass more than just an accompanist.
Yes you can. As I mentioned in my comment, one member of Ned's Atomic Dustbin pretty much strums the bass through all of every song. He is in the foreground playing in this video:
For another example If you don't mind Nu-Metal and growling vocals then check out Mudvaynes LD 50 album. Ryan Martinie uses slap, strumming, picking almost any bass playing technique you can think of.
Les Claypool is another great bassist with a nice strum technique that he uses often... Checkout Groundhog Day by Primus:
