The shell expands * only if un-quoted, any quoting stops expansion by the shell.

Also, a brace expansion needs to be unquoted to be expanded by the shell.

This work (lets use echo to see what the shell does):

$ echo *.{ext1,ext2}
a.ext1 b.ext1 a.ext2 b.ext2

Even if there are files with some other names:

$ touch {a,b}.{ext1,ext2} {c,d}.{ext3,ext4} none
ls
a.ext1  a.ext2  b.ext1  b.ext2  c.ext3  c.ext4  d.ext3  d.ext4  none

$ echo *.{ext1,ext2}
a.ext1 b.ext1 a.ext2 b.ext2

Why that works?

It is important that we understand why that works. It is because of the order of expansion. First the "Brace expansion" and later (the last one) "Pathname Expansion" (a.k.a glob-expansion).

Brace --> Parameter (variable) --> Pathname

We can turn off "Pathname expansion" for a moment:

$ set -f
$ echo *.{ext1,ext2}
*.ext1 *.ext2

The "Pathname Expansion" receives two arguments: *.ext1 and *.ext2.

$ set +f
$ echo *.{ext1,ext2}
a.ext1 b.ext1 a.ext2 b.ext2

The problem is that we can not use a variable for the brace expansion.
It has been explained many times before for using a variable inside a "Brace Expansion"

To expand a "Brace Expansion" that is the result of a "Variable Expansion", you need to re-submit the command line to the shell with eval.

$ list={ext1,ext2}
$ eval echo '*.'"$list"

Brace --> Variable --> Glob || --> Brace --> Variable --> Glob
........ quoted here -->^^^^^^ || eval ^^^^^^^^^^^^^^^^^^^^^^^^^

Values of the file names bring no execution problem for eval:

$ touch 'a;date;.ext1'
eval echo '*.'"$list"
a;date;.ext1 a.ext1 b.ext1 a.ext2 b.ext2

But the value of $list could be unsafe. However, the value of $list is set by the script writer. The script writer is in control of eval: Just not use externally set values for $list. Try This:

#!/bin/bash
touch {a,b,c}.ext{1,2}
list=ext{1,2}
eval ls -l -- '*.'"$list"

A better alternative.

An alternative (without eval) is to use Bash "Extended Patterns":

#!/bin/bash
shopt -s extglob
list='@(ext1|ext2)'
ls -- *.$list

Note: Please be aware that both solutions (eval and patterns) (as written) are safe for filenames with spaces or new lines. But will fail for a $list with spaces, because $list is unquoted or the eval removes the quotes.

echo prints its arguments separated by spaces, even if they include (or generate) newline characters. Additionally it adds one newline character at the end of its output (unless it's echo -n).

Use printf:

printf '%s\n' {0..4..2}" "{0..2..2}

When echo does something unexpected, always consider printf. After you get familiar with printf it may even become your first choice.