If I want to start my own full node, is it better to simply let the client update on its own?
If you actually mean "download whole blockchain" when you say "update", yes. Although you may want to make few custom configuration (e.g. use more RAM) to speed up the process.
Or is there somewhere I could download a trustworthy bootstrap?
No one make such file these days.
Would the bootstrap save any time in downloading?
No. In fact it'd slow down sync process[1-2].
I don't feel like running my node if this situation doesn't improve. Using my limited resources to benefit a bunch of spammers? No thanks.
Spammer don't care about amount of Bitcoin full node though. Besides, you still get benefit when you have privacy concern, serious security concern or want to run specific self-hosted application (e.g. block explorer or Electrum server).
[1]
https://bitcointalk.org/index.php?topic=986676.msg10753296#msg10753296[2]
https://bitcointalk.org/index.php?topic=2234542.msg22524484#msg22524484