Though now a days the semi-modal (Quasi-modal) use of 'Need' has fallen out of favour, it still smacks at times of its modal nature : The expression of subjective attitude and opinion for possibility, necessity or contingency. So, we may well argue that we use main verb 'Need' as an alternative to semi-modal 'Need' exclusively used in negative sentences and formal English.
Both these examples use 'need' as main verb but in the latter carrots are the passive recipient of someone's necessities (here 'need' imbibes its modal/ semi-modal nature)
Cambridge.org says we must use the main verb "Need" when it is followed by a noun phrase or -ing clause (the naming of a state , no direct activity involved of the subject) as shown in the above examples.
Again, the main verb "Need" is followed by 'to' when used with another verb.
- I need to have my hair cut.
It is a prompting where I may not be the doer necessitating passive form of the infinitive used. Once we accept example (1) in the post, example no. (3) is struck off by the same logic because carrots don't desire to chop themselves.
We have already said as a main verb 'need' requires gerundial form or a noun phrase. What remains to be seen as to why we must also discard example no.(4).
A Participle phrase that is not absolute always refers to the subject in the main clause. In the absence of any such reference such participle phrases are meaningless. Let us attempt to put the carrots to a reasonable use:
- Being chopped the carrots need to be washed. Isn't it!
The claim made in the post "need" makes use of active gerund- participal clause, is not well founded. It is as passive as is the overt passive infinitival clause in the first example.
To be chopped
Chopping by somebody.
In both the examples no.(1) & (2) the modal nature of NEED, call it semi-modal if you so like, comes to the fore (just an external necessity to which carrots are dumb spectators).