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And is it possible to sync to the current Algorand state by downloading and verifying the complete transaction history, like I can with Bitcoin? And if so, how big is that download?
Bitcoin limits the tx rate on purpose so that the Initial Block Download remains feasible on average hardware, and no trust is needed beyond the integrity of the client software.
1.) decrease mining difficulty, i.e., the amount of hashing required to produce a valid proof-of-work on average. This impacts the security of the network.
2.) increase the block size so you can fit more transactions into a single block. This negatively impacts network latency (and therefore scalability, as it increases the burden of propagating messages across the network).
To your question on Algorand: yes, new consensus participants can quickly and easily sync with the current state of the blockchain through fast catch-up [1], without any meaningful security degradation [2].
it seems like that's exactly what you want. Given that you are spreading lots of misinformation and don't seem to be trying to understand the other side's point.