What part of the US do you live in?
(I only ask because, if you're on the US west coast, it's much faster and more efficient to cross the Pacific to get to Thailand; if you live on the east coast, going west isn't terribly more efficient, but still may be slightly so; as a west coast resident who to goes to Thailand fairly often, it seems completely crazy to me to fly to Europe first).
Overall: Personally, I would never do what you are suggesting, and it seems insane to me.
You're going all the way from the USA to Thailand and back. Exactly how long is your trip? (It's hard to tell, since you're speaking in vague terms about X nights here, Y nights there). No matter which way you go, you will burn the equivalent of at least 2-4 days just getting there and back, on planes and in transit -- and that's if you are efficient. By going via Europe, you add to even more time that you lose in transit (don't just add up the flight hours -- you need to account for all the wasted time getting to/from each airport, time lost futzing with the logistics of ground transport, hotels, etc., and the exhaustion of long flights). After you subtract all that from your usable time in SE Asia, then you lop off FOUR more days for Rome...seriously?
On a short trip (and to be honest, for a trip to Southeast Asia, your trip is quite short) with a side trip to Italy on your way, you've burned far too many days, you end up with, what, maybe a week in Thailand? Maybe less? That's insane...I mean, unless you are lucky enough to be able to travel to SE Asia very often and you can/will go back many, many times and very soon. I actually do travel to SE Asia fairly often (every year or two, in non-pandemic times). Still, it's a long, long way from home, and for me, it does not make sense to go all that way to the other side of the planet for a short amount of time there.
Unless you are able to fly between the USA and SE Asia so frequently that you can go back any time, I think it makes sense to maximize your time spent there. You get more "bang for your buck" (the monetary cost of the flights, the opportunity cost of taking weeks off away from home, both limit most people). Generally speaking, I won't cross the Pacific unless I will have an absolute minimum of 2 weeks on the ground there (not counting travel time).
Of course, only you can decide what's right for you. But for me, it would be madness to carve 4-6 days for Rome out of an already-too-short trip to Thailand. My suggestions is to look at your plan, you count your days carefully (and honestly), and see how many full days --not counting travel days -- you would have at your primary destination. There are a lot of worthwhile things to see/do in/around Thailand, and it would be painful (to me at least) to go all that way and end up with a very short time there limiting what you can do/where you can go.
Hope that helps. Good luck.