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I do have an application somewhere which does this (it's called HTMLShrinker). I downloaded it ages ago, found it to be quite clever but never had any reason to use it. As I recall, it automatically generates backups of the originals for you.
There is one potential trap you have to watch out for. Suppose you have text links at the bottom of your page that look like this:
Home Sales Order Contact
...and you've written the <a> tags like this:
<a href="index.html">Home</a>
<a href="sales.html">Sales</a>
<a href="order.html">Order</a>
<a href="contact.html">Contact</a>
(one on each line). That's okay, because the line breaks are converted into spaces. But if your application (like my HTMLShrinker) removes line breaks completely and only converts multiple spaces or tabs into (single) spaces, your links will look like this:
HomeSalesOrderContact
...not what you want. I got around that by putting a space at the beginning of each line if I didn't want it to run straight into the previous line (and, although I have never used HTMLShrinker on a published website, I haven't got out of that habit!). Your application might have been programmed more carefully, of course.