(I don't know if this helps but just for your information...) In Japanese schools we are taught this way:

たい is a 形容詞型助動詞 (an auxiliary that conjugates like an i-adjective).

It conjugates to たく, たかっ*, とう in 連用形 (continuative forms).

So 行きたくない consists of:

行き -- 連用形 of the verb 行く
たく -- 連用形 of the auxiliary たい
ない -- 終止形 (terminal form) of the 補助形容詞 (subsidiary adjective) ない

And 行きたかった consists of:

行き -- 連用形 of the verb 行く
たかっ -- 連用形 of the auxiliary たい
た -- 終止形 of the 過去の助動詞 (past tense auxiliary) た

So 行く is a verb and acts as a verb, but 行きたい acts as an i-adjective and conjugates to 行きたく, 行きたかっ, 行きたけれ, etc. (but it's not that 行きたい is an i-adjective.)

_{*たかっ was originally たかり. たかり(+た/て) changed to たかっ(+た/て) due to 促音便.}

(I don't know if this helps but just for your information...) In Japanese schools we are taught this way:

たい is a 形容詞型助動詞 (an auxiliary that conjugates like an i-adjective).

It conjugates to たく, たかっ*, とう in 連用形 (continuative forms).

So 行きたくない consists of:

行き -- 連用形 of the verb 行く
たく -- 連用形 of the auxiliary たい
ない -- 終止形 (terminal form) of the 補助形容詞 (subsidiary adjective) ない

And 行きたかった consists of:

行き -- 連用形 of the verb 行く
たかっ -- 連用形 of the auxiliary たい
た -- 終止形 of the 過去の助動詞 (past tense auxiliary) た

So 行く is a verb and acts as a verb, but 行きたい acts as an i-adjective and conjugates to 行きたく, 行きたかっ, 行きたけれ, etc. (but it's not that 行きたい is an i-adjective.)

_{*たかっ was originally たかり. たかり(+た) changed to たかっ(+た) due to 促音便.}

(I don't know if this helps but just for your information...) In Japanese schools we are taught this way:

たい is a 形容詞型助動詞 (an auxiliary that conjugates like an i-adjective).

It conjugates to たく, たかっ*, とう in 連用形 (continuative forms).

So 行きたくない consists of:

行き -- 連用形 of the verb 行く
たく -- 連用形 of the auxiliary たい
ない -- 終止形 (terminal form) of the 補助形容詞 (subsidiary adjective) ない

And 行きたかった consists of:

行き -- 連用形 of the verb 行く
たかっ -- 連用形 of the auxiliary たい
た -- 終止形 of the 過去の助動詞 (past tense auxiliary) た

So 行く is a verb and acts as a verb, but 行きたい acts as an i-adjective and conjugates to 行きたい, 行きたかっ, 行きたけれ, etc.

_{*たかっ was originally たかり. たかり(+た/て) changed to たかっ(+た/て) due to 促音便.}

(I don't know if this helps but just for your information...) In Japanese schools we are taught this way:

たい is a 形容詞型助動詞 (an auxiliary that conjugates like an i-adjective).

It conjugates to たく, たかっ*, とう in 連用形 (continuative forms).

So 行きたくない consists of:

行き -- 連用形 of the verb 行く
たく -- 連用形 of the auxiliary たい
ない -- 終止形 (terminal form) of the 補助形容詞 (subsidiary adjective) ない

And 行きたかった consists of:

行き -- 連用形 of the verb 行く
たかっ -- 連用形 of the auxiliary たい
た -- 終止形 of the 過去の助動詞 (past tense auxiliary) た

So 行く is a verb and acts as a verb, but 行きたい acts as an i-adjective and conjugates to 行きたく, 行きたかっ, 行きたけれ, etc. (but it's not that 行きたい is an i-adjective.)

_{*たかっ was originally たかり. たかり(+た/て) changed to たかっ(+た/て) due to 促音便.}