The Clans would also probably look at an untouched (from the stupidity of the current setting's factions, I mean) and more technologically advanced Modern Earth ("Wait, if we help them they can build a battlemech that's technologically superior to anything we have right now? In just a few years? I'm throwing c-bills at the vid-screen, but nothing's happening!") as some sort of utopia/promised land, too. They wouldn't want to destroy it or do any significant damage to its population or capabilities; rather, use them to advance their own stuff or even recreate it from scratch as something better.Probably fail utterly the first time, due to the pirates getting in and out faster than the politicians have come to grips with what is happening and sent any military assets. By the time the pirates return for another bite six months to a year later, there's likely to be a doctrine to deal with them. My first guess would be an overwhelming aircraft/cruise missile spam hitting their 'mechs and possibly DropShips with copious amounts of explosives.
The inverse of the pirates. Since the Combine won't grab and run, but instead will send possibly a 'mech company. They'll also open talks to arrange for a surrender, and thus sit there long enough for a diplomatic envoy to show up. Relations are going to break down pretty quickly since Combine's gonna Combine. The 'mechs will likely rampage once it's clear Earth isn't going to roll over, followed by cruise missile and aircraft bombardment taking the 'mech company down.
However, the second response from the Combine will be much harsher. Earth will likely get at least a Sword of Light dropped on them, and the Combine is in a position to take up orbit and plan their attacks carefully, lifting back off the ground rapidly as needed and sending ASF bombardments to soften up and destroy offending missile batteries. The timeline is a touch dicey, in 2900 the Combine still had a few WarShips left, probably not enough that they'd peel one off just to do an orbital bombardment, but maybe they would, especially if they look at how much Industry Earth has.
It's worth noting that the early clan invaders were not the caricatures they became later, they were quite keen on maintaining stealth, gathering intel, and generally being sneaky gits during their initial phase of invasion. The whole opening up with a Batchall on day 1 came later.
They also both absolutely have WarShips, and have Elementals which will give them an infantry-like force that's tougher than a Tank and hits harder too without being a walking skyscraper for all to see and paste with missiles.
The upshot of this is that the clans have a lot more options in their toolbox than the Combine or the Pirates and can select a much more measured response. The 3050 clans have also not yet become idiots who look at five million infantry and a hundred thousand tanks, then bid a trinary of 'mechs in a frontal because that just means the victory will be all the more glorious.
This means the Clans have the best chance of taking Earth by far, though that would have some hilarious knock-on effects down the line because you don't add eight billion people to your relatively small population without them irreversibly altering your culture and ways into their own. It would be the Mongols conquering China all over again.
Hell, I can imagine another political faction emerging distinct from the Wardens and Crusaders that'd want to use Modern Earth to create a new Star League from scratch than trying to build one on the bones of Inner-Sphere -- some clans, like Goliath Scorpion, I could see going all in on this.
I mean, they're not stupid (until FASA made them that way).
A darker turn for the Clans, however, could be if the remnants of Wolverine/the "Minnesota Tribe" fled into the deep, Deep Periphery, found Modern Earth, and decided to basically uplift us from the shadows so we'd eventually fuck over both the Clans and the Inner-Sphere (or, they'd go public and work with us openly).
The more we'd work on re-creating better Battletech stuff, the more Wolverine becomes more and more sure they made the right decision.