There would appear to be a hereditary component - for instance, in the TV show Jenny was considered by the Doctor to potentially be able to regenerate and she was half-Gallifreyan. If it was solely technological, this would not even be considered.
This happens again in "Big Finish" with speculation on whether Alex, Susan Foreman's half-human son via David, would have inherited the ability to regenerate from his mother - which he hadn't, but again, it was openly considered to be a possibility.
There's also Melody - who is not even Gallifreyan but "inherited" the ability to regenerate, thanks to her being conceived whilst in the untampered Schism.
Exposure to which, within episode, is attributed to being why the ability to regenerate formed within the Doctor's people.
There's a book too, which cites it being in relation to a "triple helix" - which could in turn, have developed in relation to the Time Vortex, so it doesn't go against any of the above.
Along with this, Big Finish detailed the "Dogma Virus" which broke out during the Time War - something triggered by regeneration that would essentially turn it's victims into mindless, violent "zombies". Mattias had to work to quarantine it while Romana was away as it threatened to spread throughout all of Gallifrey - the implication being that all Gallifreyans are susceptible to it a quarantine is not enforced: which by necessity, this being regeneration linked, has to mean that they can all regenerate too.
Then there's also this very explicit tidbit here:
"One Gallifreyan wrote that regeneration was a natural process for all
Gallifreyans, and was not exclusive to the Time Lords."
(in regards to "A brief history of Time Lords")
This also goes along with "Time Lord" being a rank trait and not a biological one.
A Note: It's potentially true that this started off as a technological trait, one introduced to Time Lords as some other materials would suggest, but that in turn eventually spread and became a natural/hereditary one over time.
As for the daleks shooting and killing them: People who can regenerate can die if just killed fast enough or they've ran out of regenerations. They need time to kick start the process, which daleks do not tend to often provide.