Where are you getting this data to conclude we haven't seen a such a spike?
https://www.cdc.gov/nchs/nvss/vsrr/covid19/excess_deaths.htm
* Select "Excess deaths with and without COVID-19" for the dashboard
* Enter "New York City" as the location (New York state as a whole shows a similar trend)
New York state was locked down March 20th. Notice the marked increase in non-COVID-19 related deaths starting March 21st, increasing steadily into April (almost 200% higher than predicted number of deaths). The tooltips also say that this data is still incomplete, so these numbers could climb.
Now maybe it's unrelated, or maybe it's COVID-19 related that was incorrectly assigned, but you seem fairly confident that it's unrelated to the other factors the OP mentioned. I don't see how you can reach that conclusion given the data available at this time.
If you do it in New York, chances are you would be attributing COVID-19-related but unconfirmed deaths to lockdown.