The idea of covering the paint with some other kind of coating doesn't sit well with me. No matter what you put over that paint, it's only going to stay down as long as the paint does. Once that paint starts to let go, that's the beginning of the end of your repair.

I would remove the paint first, and stripping it off is as good an option as any (although sand blasting it off would be faster and cleaner, it would undoubtedly put more lead dust in the air.)

If it were me, I would apply a thick coat of paint stripper,
cover with wax paper to prevent the stripper from evaporating, and then scrape the softened paint off with a Nestor Scraper, named after it's inventor:

You make a Nestor Scraper by gripping a single edge razor blade:



in a pair of needle nose style locking pliers:



The resulting tool holds the razor blade at a near perfect angle for scraping. Deposit the paint you shave off in a soup can or something you'll be throwing away anyhow.

If your steps are wood, dull the edge of the razor on a belt sander first to prevent the razor from cutting into the wood.

The thin edge of the razor will cut into the softened paint fairly easily making removal faster.

Once the paint is off, then you can put whatever you want on the steps, including indoor/outdoor carpeting.