Is it plausible that strict lockdowns made it more likely for the new variant of COVID to have emerged?
While I get your intuition, the hypothesis seems implausible

Emerge of new CoV variants should be considered proportional to the
current spread of the virus. Each virus has a certain mutation rate that surely does not depend on lock-downs.
It seems implausible to interpret
lock-downs as physical limitations to viral spread, that could potentially be
overcome by any mutation.
Even if CoV magically learned to
teleport to different households, such an ability would always represent immense advantages; an advantage that does not need lock-downs to be effective
Lastly, the differential viral competition is neglectable so far; we are at 6 of 100 thousand infected persons. So basically a 0% chance that a virus would infect a person who has acquired an immunity due to a previous CoV infection. In your model this person, who is immune, further would need to NOT be immune because quarantine prevented the (previous) infection.