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I very briefly played Rolling Thunder 2 recently as well, while going through my Mega Everdrive collection. It's a damn good looking game, at the very least, but yeah not being able to shoot while jumping is really weird if you're used to Shinobi. I want to put some more time into it soon

Ouch! Poor man's Shinobi indeed.
No, I can see why it would feel weird for anyone new to the game, but to be honest it's something I don't even notice anymore.
RT feels a much more deliberate and methodical game that Shinobi's freestyle hi-speed action.
Both classics, but I've always been more in the RT camp.

You can always try the MD/Gen-exclusive, Rolling Thunder 3. It has a faster, looser feel to it, partly because of things like being able to shoot while jumping.

Except Rolling Thunder came first. :p If anything, Shinobi was the one ripping off Rolling Thunder (although the two games have pretty substantial differences). Second on the RT3 recommendation. There's also Code Name: Viper on the NES, which was apparently handled by the same team that ported RT1 to the NES (Arc System Works back when they were doing outsourced work for other companies).

Cleared Bayou Billy and TMNT2 on the NES, both using continues. I tried to do a no-continue run for both games, but the driving stages in Bayou Billy are way too frustrating with their cheap one-hit deaths, while TMNT2 simply drags on way too long for me. Neither game are particularly interesting enough for me to attempt a no-miss clear, although I do think Bayou Billy's flaws are sometimes blown out of proportions. While Mad City (the Japanese version) is a much easier game, it's not necessarily better for it (the fighting mechanics in both versions are pretty simplistic either way). In some ways I actually prefer the difficulty level of Bayou Billy over Mad City, since I remember Mad City being way too easy for my liking. Plus, I prefer Annabelle's wardrobe in the NES version.

Speaking of Shinobi AC and RT : what do people in here think of E-Swat?

Not as good as either of the games you mentioned.

I've not played it in forever, but I remember thinking it was a bit bland, and not a path on the MD version.