What LBS man says is tosh. I do 30-50 miles a week on my 1980 Raleigh singlespeed and over the summer I did three times that. The tubing in a classic bike is far more resistant to rust than the metal in a classic car (I should know, I've had a few of each) so unless you pick something out of the lake it should be alright (most old bikes have surface rust which is nothing to worry about, but you'll be able to spot serious rust anyway). You won't notice a difference in braking unless you're hurtling about at 25-30mph, in which case I'd recommend upgrading to modern calipers and rims.
What's more, my Raleigh cost £12 on ebay. The singlespeed conversion cost me £8 for a BMX freewheel, £16 for a bottom bracket, and £60 for a pair of cheap 700c wheels, £20 for tyres and tubes. That's literally all the 'necessary' mechanical stuff to get it where it is today (I could've kept the original 27" steel wheels too but they were a bit too heavy for my liking). So £36 for a perfectly good singlespeed, or £116 with the modern lightweight wheels and tyres which made it much more rideable.
Don't fall into the same trap I did though - I liked it so much I resprayed it, gave it a funky red chain, stubby mudguard, Brooks saddle, two sets of bar tape and tyres (first both red, then brooks leather tape and classic gumwall tyres) and I'm thinking of getting a new chainset too! Upgrading your bike is bloody addicting
but ultimately very satisfying.