Proponents of mandatory voting also tend to be supportive of other voting reforms which would make it faster and easier to vote. For example, vote by mail and removing the selective service requirement for men. With some properly implemented reforms, the time it takes to vote could easily be reduced to ten minutes or less.

Mandatory voting would also push the responsibility of ensuring that people have an opportunity to vote on the government, which is really how it should be anyway, but it behooves the powers that be to keep turnout lower. At least in theory, this would obviously need to be codified.

As for religious exemption, I think most mandatory voting advocates would only want to require that every citizen turn in a ballot, but not that it be filled out in any particular way. An objector could turn in a blank ballot or write in a fictitious candidate by that standard. They would have no real sway on the political state, so unless they have a religious objection to filing paperwork they don't have much to complain about. Even so, there could easily be a way to allow people to apply for an exemption.

Your right about punishments being a fraught subject here, though I think everyone's on the same page about them being pretty light. A "realistic" (this whole thread is pretty unrealistic) implementation would probably involve some minor penalty on your tax returns, though personally I'm not happy with that solution.

The point is to push the onus of providing voting opportunities on the government, and increase overall turnout. As I can anecdotally attest, and as you seem to have found on your own, people who don't vote often do have strong opinions. They either don't vote out of laziness or a lack of access. Mandatory voting would fix the former and would necessarily be bundled with legislation to fix the latter.