Just going through the troubleshooting list, apologies if you've tried these already. I'll use Abbey Road Two as my example, but the process is the same for Mellotron V.
When adding a track to Logic, choose a MIDI instrument track. Click the instrument slot, then mouse over 'AU Instruments', then 'Spitfire Audio', then click 'Abbey Road Two'. You can also, in one of the newest updates, simply do a search in the instrument slot. If you can't find it, then...
Go to the 'Logic Pro' menu in the top left, then mouse to 'Settings' and click 'Plug-in Manager'. You can scroll through the 'Show All' list until you find it, look through the 'Manufacturer' list, or do a search for the instrument in here. If you locate the plugin and it says 'couldn't be opened' or 'failed validation' in the 'Compatibility' column, select it and click the 'Reset & Rescan Selection' button. If you don't find it, click 'Full Audio Unit Reset' and see if it shows up. If it still doesn't show up, then...
In Finder, click the 'Go' menu at the top and select 'Go to Folder'. In the dialog that pops up, type in (or copy/paste) "/Library/Audio/Plug-Ins/Components" without the surrounding quotation marks. In the folder that comes up, ensure there is a file called 'Abbey Road Two.component'. If it's not there, then...
Open the Spitfire Audio app and click on the Abbey Road Two icon. Ensure that all the listed downloads show as installed. If so, click the 'repair' button next to each download.
Regarding the slowness of the Arturia Software Center, they released the next version of their very popular Pigments soft synth today. It's very possible that their servers are taking a huge hit currently.