2001 Ford Escape Cold start problems


Hoping someone will be able to solve mystery that has mechanic stumped and me tearing my hair out. Have 2001 escape v6, 130kms. About 3 weeks ago, coolant light would come on when starting cold, then go off after a couple of minutes. One afternoon, car would start, then engine die when trying to shift into gear. Roadside assistance found a collapsed air hose, replaced, and car started. Took to mechanic and had full service. Then, for the next day coolant light started coming on and off - back to mechanic who said they hadn’t bled the coolant lines enough - topped off the coolant. Now, at cold start it takes 6 or more attempts to get the engine to stay running. If it parked for less than 3 hours, starts fine first time. Mechanic says they’ve checked all the vacuum hoses, and topped off the coolant again

I have had some success turning the key to ON 5 or more times before trying to start it - so had the fuel pump replaced on theory it was a check flow valve problem. But it actually seems worse now. It’s not showing any codes, so have no idea what to do now.
Any help would be so appreciated - can’t keep not knowing if the car will start when I’m driving little kids around