XD dawwww if he asked she might say yes but she'd set up a separate office for him with Martha next door and Jack's office between them as INTERMEDIARY NESS but then he could do inventing and things and she would test his stuff :3
She's more worried about him causing international incidents. XD IE: Sycorax, Harriet Jones, that cult with the Silence, the TimeLords, THE PANDORICA, the time he tired to sell his memories :V
I think Amy's talk with Rose about the actual function of Torcwhood and why the Doctor(s) is not its biggest fan is a big reason she would be a lot more like "okay" to 10 asking her to join because
she'd be touched that he trusts her and/or at least wants to make sure she's on the up and up and not disappointing him, and + it's like what she was gonna tell Amy that it's for people who need the work as part of their identity/sanity tool
Sarah is going to be from about 1981, so a year after everything but no K9 or adventures on her own yet. and she hasn't met doctors 6-11 yet xD but she'd be able to tell the Doctor right away
he's given her plenty of positions alreadylook now someone app Ace and or Bill