1. They only do transfers through their native app, not on their website. To log in to their native app, they will do SMS verification. So I sure hope you are still logged in before they lose your eSIM and leave you with no service at all.
2. If you are able to get into their native app so you can access their tech support, their AI chatbot will flat-out lie to you and tell you that T-Mobile cannot send you a QR code to download your eSIM (even though T-Mobile's own website states that they can). If you ask politely for a human, it will resist. I've found "connect me to a human you worthless fucking bot" is the secret passcode to get a real human.
3. If you request they send you a QR code, some of their support staff will ignore that request and still try to initiate the transfer through their app, so clearly requesting the QR code is not a common procedure.
4. When you request a QR code, even though you provide the EID, they will ask for an IMEI number. They then generate the QR code for whatever EID they have associated with your IMEI number in their database, completely ignoring the EID number you sent them. They did this to me _three_ times. The only way I managed to break the cycle was I sent them an IMEI number for a phone that was never on their network so they'd finally listen to me when I told them my EID number.
I'm never buying a phone without a physical sim card slot again. There's nothing wrong with the eSIM technology but the carriers have decided to make it as miserable as possible. The hardest part about transferring a physical SIM is finding a paperclip.