If you can solder, then you could try to fix the contacts.
From the looks of that motherboard on eBay, you will have to taker the motherboard out.
According to [https://www02.cp-static.com/objects/pdf/7/74c/1349624249_1_voedingadapters-en-omvormers-lenovo-65w-3pin-45n0497.pdf|T440s fru_list], SIM tray is a separate part(Its FRU 04X5345, description is “SIM Tray”), but it isn't CRU (customer replaceable unit) in the parts_list.
I suspect that Lenovo won't/can't do it because their repair facilities don't have the necessary equipment and skills. You may find a local PC repair shop or similar facility that specializes in doing such precision work.
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While the problem came to SIM card reader, you should be careful as it is a more precision work.
If you can solder, then you could try to fix the contacts.
From the looks of that motherboard on eBay, you will have to taker the motherboard out.
According to [https://www02.cp-static.com/objects/pdf/7/74c/1349624249_1_voedingadapters-en-omvormers-lenovo-65w-3pin-45n0497.pdf|T440s fru_list], SIM tray is a separate part(Its FRU 04X5345, description is “SIM Tray”), but it isn't CRU (customer replaceable unit) in the parts_list.
I suspect that Lenovo won't/can't do it because their repair facilities don't have the necessary equipment and skills. You may find a local PC repair shop or similar facility that specializes in doing such precision work.