As salaam alaii kumm,
Deobandi Ulamahs frequently quoting zaeef hadiths is a notion spread by the La Madhabis, this is one of their's nice simple dirty old trick. They have made it so popular that anything from Deoband is zaeef (weak).
For ex., I had a close relative of mine visit me and I showed him my small collection of books, he didn't touch them, he did not even see them (titles) and in less than a minute, said "bro you buy books from so and so publishing house, all their books quote strong Hadiths, don't read these" he was aware that I was pro (India) Deoband. This relative was frequenting Al Maghrib, his zaahir is not of a Muslim, I don't know of his baatin, and after some time, he was talking against another book. May Allah ta'ala guide him. Aameen. (His behavior when he saw the books was like as if he saw snakes in the zoo, he saw them and instantly stepped back as if there was an open manhole)
Point: The laymen are being brainwashed, a lie is being spread by the Ghair Muqallids.
And even if we accept that Deoband ulamah's frequently quote zaeef (or fabricated) Ahadiths, yet relatively there is more good happening through them then the strong Hadiths (acc. to them only) quoting La Madhabis.
I'm sorry for sounding aggressive.


? Should the person still keep quoting the Hadith?