Is it sensible to buy quality old saws and to have them profesisionally sharpened?
No!
However, it's not the new/old debate that I'd take issue with. It's the professional sharpening bit.
As with most things to do with hand tool woodworking, the key to getting the most out of the tools is maintaining them well; and the key to that is usually keeping them sharp.
If you want to get the best out of traditional panel and back saws, you need to learn to sharpen them.
Learning to sharpen a saw does not just result in you having a sharp saw that cuts straight, but it is also the best way to gain appreciation and understanding of how teeth angles and offset affect the way the saw operates.
And the best thing about it is that it's not particularly difficult. You can get a lot done with a baton with a slot cut down its length, a vice and a triangular file.
I'd suggest starting with a second hand 7tpi rip panel saw. The Aussie Woodshed chap is producing some interesting starter videos around saw sharpening which I think would be a good place to start for information about how to sharpen:
I'd start by simply sharpening the teeth for rip. Just taking an old rip saw, making some cuts, and then sharpening the teeth and sawing again can be a really satisfying experience.
The simplest sharpen is rip with the side of the tooth facing forward vertically. That means holding the triangular file such that the file side facing the saw handle is vertical (with the top edge of the saw held horizontal in a vice)
Then try sharpening a crosscut saw. That's more complicated because the teeth need to be cut at alternating angles and with more of a slope than rip teeth. But it really isn't that hard.
Then go on to settings the teeth. With a sawset it really isn't difficult. More repetitious and fiddly than anything else.
If you then look at jointing (levelling the top of the teeth before sharpening) you'll have the basic skills.
Once you've got the basics you can start experimenting with different configurations. I think you'll then appreciate the benefit of having a number of saws configured for different cuts.
Then I think you'll get the best out of the hand saw experience
And one last thing, getting a good table lamp and some magnifying glasses can make all the difference. Keeping track of which tooth is next to be filed is a nightmare if you can't see what you're doing.