Should a Brahmin or a normal person strictly follow vedas to attain good birth or to attain Moksha.
From purely tantric perspective:
No, not necessarily. A person can be a 'Pashu', 'veer' or 'divya' - 'sadhaka' accordingly. Although there is a general agreement inside tantras that 'VedicAchara' is followed by 'Pashus' (Also referred as 'PashvaAchar' पश्वाचार ).
'Shkta' are of two types, one are called the 'DakshinAcharShakta' other one are the 'VamaAcharShakta'. 'DakshinAchar' says to act according to 'SwaDhrmha' i.e. the Varna one is born into.
निरुत्तरे-- द्विजातीनाञ्च सर्वेषां द्विधा विधिरिहोच्यते । दिवा च पाशवं कर्म रात्रौ कर्म च कौलिकम् ॥ १२ ॥
Any Dvija (Twice born) has two things to follow, in the day he performs the puja according to PashvaAchar (पश्वाचार), and in the Night practice the Kaulachara.
(Sarvollāsatantram, 18th Ulhassa)