Philosophy has not advanced very far since Ancient Greece and Rome
Why on earth do you think that? What about contemporary metaphysics, logic, political philosophy?
and every philosophical point of view, when extended, becomes absurd.
Isn't that a philosophical point itself? In all seriousness though, it might not be a good idea to posit this in general without very specific reasons as to why it can be said generally. Depends on what you mean by absurd. Also, why's absurdity itself an issue?
Meanwhile, in little more than 3 centuries, and mostly in the latter
half of that, Science has transformed (and multiplied) the lives of
virtually everyone on the planet, although not all for the better.
It's also given us weapons to decimate thousands of people in an instant, or given us technological advances which makes us able to us millions of animals for our own gain without necessarily caring about what happens to them. Don't take this the wrong way: I only want to look at all the results.
I suppose you could argue that Science and technology is a branch of
philosophy, but if so, it is probably the only branch worth having.
(1) Only science produces good results.
(2) Only what produces good results is useful.
(3) We ought to only keep whatever is useful.
(C) We only ought to keep science.
Premise 1 is flawed, more on that below. Premise 2 rests on philosophy, also more below. Premise 3 might potentially lead on into reductio ad absurdum.
Philosophy may be a pleasurable pastime, but it doesn't really solve
real world problems.
This is wrong on multiple levels. First of all, to even be sure what the "real world problems" are that we ought to solve, ethics kind of does come in handy.
(If we want to argue that it isn't, well, we're also already doing philosophy.)
We could also see philosophy as keeping our "solutions" in check. Otherwise we might blindly follow ideologies when making our solutions. Especially continental philosophy often takes on such a role.
We can also see other affairs as taking such a role. Criticism of science for example can have the aim of improving it. Even people like Bruno Latour think this. But it doesn't even have to be criticism at all! Relevant quote by Dennett: "There is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination."
Surely, advances on logic since Aristotles haven't been useless.
On example of how relatively modern philosophy still provides foundations for science: Gödel was a huge influence for Turing, bringing us to Computer Science. We could trace Gödel back to Russell, to Frege, ...
Some political philosophy is a HUGE deal. It'd be a mistake to assume that f.e. Rawls' theory doesn't have influence over political discourse and decisions.
Furthermore, some branches of science don't exactly have "practical relevance" either. Does this mean why should try to get rid of them? If so, how do we do this without already engaging in philosophy?