If Buddhas don't feel bliss, what is the purpose of attaining nirvana? If I have zero feelings won't I be a dead robot? And how would I have love and compassion for beings without feeling?
From AN 9.34:

Ven: Sariputta: “Reverends, extinguishment (Nibbana) is bliss!
Ven. Udayi: “But Reverend Sāriputta, what’s blissful about it, since
nothing is felt?”
Ven. Sariputta: “The fact that nothing is felt is precisely what’s
blissful about it.

Sukha or happiness for an unenlightened person is experienced when encountering pleasant feelings (from the six senses, including thoughts and the intellect) or when encountering the cessation of painful feelings (from the six senses).
But for a Buddha or arahat, sukha or bliss (in this context) is experienced when encountering neutral feelings, no feelings and Nibbana.
Please see this answer for commentaries by Nyanaponika Thera. The supporting suttas are MN 44, SN 36.5 and AN 9.34.
In addition, the Buddha and arahats can experience emotions or states of mind which are not tainted with the three poisons of greed (lobha), aversion (dosa) and delusion (moha). This includes the four brahmaviharas of loving kindness (metta), compassion (karuna), empathetic joy (mudita) and equanimity (upekkha).
The brahmaviharas are considered skillful states of mind.
An enlightened person is not an emotionless person. It's simply a person who does not have cravings, does not have plans to fulfill sensual cravings, does not have plans to become something and does not become afflicted with negative states of mind that arise as a side effect of having craving. Also please see this question.
The purpose of Nirvana is permanently ending suffering - it's not about experiencing bliss.

Adding on based on a comment by Ruslan, the Buddha did have his five aggregates (form, feeling, perception, mental formations and consciousness) intact and functioning without being tainted by the three poisons, after attaining Nibbana, and could still experience pain and pleasure, but did not suffer from it. Please see this answer for details.