OK, so I'll provide a summary of all the "circumstantial evidence" for Altaic that I can think of.
-vowel harmony found in every branch except, perhaps, for Japonic
-rhotic phonemes do not begin native words in any branch
-a system of complementary distribution for velar & uvular plosives & fricatives is found in Mongolic, Tungusic, & in some Turkic languages
-SOV syntax found in every language
-all languages are agglutinative & strictly suffixing
-a case system with nominative unmarked in Turkic, Mongolic & Tungusic; a system of "particles" found in Korean & Japonic
-no grammatical gender
-no distinction of singular & plural, except for animate nouns & pronouns; the suffixes used to show plural have similarities: compare Japanese -ra with Uzbek -lar & Tuvan -ler; Japanese -tachi, Korean -deul, Kazakh -dar
As far as I know these same traits are found in Uralic.
True, but the other traits aren't. Many language families have similar traits.
I think I remember something from when I was younger about a Uralic-Altaic Link and being stunned by the connection of Finnish and Japanese...not realising at that age that it was only a theory.
That hypothesis has been completely debunked.