Actually,
V-M is still in business (tho' no longer manufacturing, only servicing) and has both the cartridges and styli. See link.
Since that turntable is now past its 40 years between servicing interval, you ought to clean and lube it.
It takes a whale of a lot of energy to blow speakers. Be sure you know that your system is electrically ok before you install a new one. Bad capacitors (I know we sound like a bad record here) are one reason.
It is more likely to be in the crossover or wiring connection than the actual speaker coil. But, if the speaker coil itself is blown, it can be rebuilt. There are two contingencies involved. One, since this is a stereo, you should have the other channel's equivalent speaker rebuilt at the same time.
Two, I prefer tube amps to solid state, who doesn't? But the difference in sound quality between a good speaker (nothing expensive, Radio Shack quality) and the best used in consoles back in the early sixties (excepting some Magnavox, pre-Astro-Sonic) is so tremendously great, I agree with your dad. Speaker sizes haven't changed, you can still get one that will fit. I'd replace the speakers, lube and clean the turntable, recap the amps and your beautiful Philco will sound as good as she looks.
Total investment should be under 100$, tops, if you do it yourself.
Keep asking questions, there are lots of folks here who know how to do these things and will be happy to help.
Please, I know I sound like a 15 year old, but could we have more photos, please? (I'd love some of the inside, too.)
Voice Of Music, V-M Audio Enthusiasts
www.thevoiceofmusic.com