Simplify complex command, put it into a variable - bash

Question

date +'%A %B %d' | sed -e 's/\(^\|[^[:digit:]]\+\)0\+\([[:digit:]]\)/\1\2/g
I like the output of the above command, which strips leading zeroes off days of the month produced by the date command, in the case of numerals less than 10. It's the only way I've thus far found of producing single digit dates from the date command's output for the day of the month, which otherwise would be 01, 02, 03, etc.
A couple of questions in this regard. Is there a more elegant way of accomplishing the stated goal of stripping off zeroes? I do know about date's %e switch and would like to use it, but with numerals 10 and greater it has the undesirable effect of losing the space between the month name and the date (so, July 2 but July10).
The second question regards the larger intended goal of arriving at such an incantation. I'm putting together a script that will scrape some data from a web page. The best way of locating the target data on the page is by searching on the current date. But the site uses only single digits for the first 9 days of the month, thus the need to strip off leading zeroes. So what's the best way of getting this complex command into a variable so I can call it within my script? Would a variable within a variable be called for here?
RESOLUTION
I'll sort of answer my own question here, though it is really input from Renaud Pacalett (below) that enabled me to resolve the matter. His input revealed to me that I'd not understood very well the man page, particularly the part where is says "date pads numeric fields with zeroes," and below that where it is written "- (hyphen) do not pad the field." Had I understood better those statements, I would have realized that there is no need for the complex sed line through which I piped the date output in the title of this posting: had I used there %-d instead of just %d there would have been no leading zeroes in front of numerals less than 10 and so no need to call sed (or tr, as suggested below by LMC) to strip them off. In light of that, the answer to the second question about putting that incantation into a variable becomes elementary: var=$(date +'%A %B %-d') is all that is needed.
I may go ahead and mark Renaud Pacalet's response as the solution since, even though I did not implement all of his suggestions into the latest incarnation of my script, it proved crucial in clarifying key requirements of the task.

Answer 1

If your date utility supports it (the one from GNU coreutils does) you can use:
date +'%A %B %-d'
The - tells date to not pad the numeric field. Demo:
$ date -d"2021/07/01" +'%A %B %-d'
Thursday July 1
Not sure I understand your second question but if you want to pass this command to a shell script (I do not really understand why you would do that), you can use the eval shell command:
$ cat foo.sh
#!/usr/bin/env bash
foo="$(eval "$1")"
echo "$foo"
$ ./foo.sh 'date -d"2021/07/01" +"%A %B %-d"'
Thursday July 1
Please pay attention to the double (") and simple (') quotes usage. And of course, you will have to add to this example script what is needed to handle errors, avoid misuses...
Note that many string comparison utilities support one form or another of extended regular expressions. So getting rid of these leading zeros or spaces can be as easy as:
grep -E 'Thursday\s+July\s+0*1' foo.txt
This would match any line of foo.txt containing
Thursday<1 or more spaces>July<1 or more spaces><0 or more zeros>1