The language of your post indicates a lot here. saying x? nope. y? nope. z? not possible. indicates that you have accepted defeat before the game even starts.
your mindset is critically important going into starcraft games. It is possible to make these builds work but it is going to take some practice. Start there.
Your phrasing is implying you're looking to do game ending damage with a push while playing mech, that's not how the matchup works, sure you can punish greedy zerg, but against a good player most pushes will be easily held off. What you want is a build that offers you safety and map control, here are some:
- Gumiho CC first double facto into 5 fast facto opening: Won't go into details, Beastyqt has a good guide on it: + Show Spoiler +
I like this build on safe maps, but be advised that it's an auto-loss versus ravagers all-in.
- Standard hellion/banshee 1/1/1 as was previously posted. As a diamond 2 meching terran, here's a variation I like. When starport is done, viking first to deny overlord scouting around your base, then go overlord hunting around the map (polt viking), next is a banshee for defensive purposes only (if you scout a roach all-in you can skip viking), and after the banshee you make a raven and regroup raven/banshee/viking. Then I use those 3 units a poking tool/creep denying squad. You can take a solo queen with one banshee + 1 or 2 turrets, but don't lose any unit. As long as you keep the banshee alive, this is your ticket to establish and secure a third base. When your third is setup, it's on you to identify the zerg composition and play accordingly. Good luck
I prefer to open with reactor factory (no marine, no reaper) with a widow mine drop followed by liberator harass. The advantage is that the opener puts presssure on Zerg without you having to dedicate much APM to the attack.
I build my 2nd and 3rd CC in my main while doing this attack and I typically have my 3rd CC built faster or at the same time as the Zerg.
The opener also hold most agressive openers from Zerg if you leave the mines at home until you can counterattack.
Obviously if you scout fast Ravagers (double gas, 1 base) you can not do this opener and should do the standard defense instead (bunker on low ground, bunker on high ground, fast tank, fast banshee).
TBH I think the standard 6 marine / 1 mine / 1 medivac into hellion banshee or hellion raven into 3cc that most pros have been doing is a great opening for either bio or mech. You can also skip the marine / mine / medivac pressure and go straight into hellion banshee, and you can always get a viking before the banshee / raven to help deny scouting.
The key is more about unit movement / positioning than anything. Hellion banshee does quite well vs 6+ queen openings since you can harass one area with banshees and another with the hellions, forcing the Zerg play to make more units to defend the hellions or risk taking damage (since they would need all but 1-2 of the queens defending the hellions to kill them before taking massive drone losses). Banshees also make you safe vs just about any ravager push and the only big remaining threat, queen / roach nydus, can either be denied with the viking or easily scouted by the lack of saturation at the third and is pretty easy to hold with scv pull + hellbats, banshees and whatever other units you have coming.
There's a GM level Terran streamer (AngrySC2) I watched who plays exclusively mech in TvZ. He opens with a Reaper expand into a 1-1-1 Reactor Hellion/Marine/Medivac timing, which allows him to take a fast third, do drop pressure and transition into a flexible build. He also builds a quick armoury behind this which allows him to morph hellbats when needed.
If he scouts Roaches and Hydralisks, he will build two additional factories and go into a tank build. If he scouts ling/bling, he generally goes for tanks then thors then turrets to thwart potential muta play.