Use smarter default for erc-server-reconnect-function

* doc/misc/erc.texi (Sample Configuration): Remove customization in
`use-package' declaration for `erc-server-reconnect-function' as well as
related language in the customization walk-through.  Do this because the
new default incorporates `erc-server-delayed-check-reconnect' behavior
for compatible connect functions.
* etc/ERC-NEWS: Announce new default for `erc-server-reconnect-function'.
* lisp/erc/erc-backend.el (erc-server-reconnect-function): Change
default to `erc-server-prefer-check-reconnect'.
(erc-server-delayed-check-reconnect): Use `process-send-string' instead
of `send-string'.
(erc--server-delayed-check-connectors): New variable.
(erc-server-prefer-check-reconnect): New function.
* test/lisp/erc/erc-scenarios-base-auto-recon.el
(erc-scenarios-base-auto-recon-unavailable)
(erc-scenarios-base-auto-recon-no-proto): Remove unnecessary
`erc-server-reconnect-function' binding because the new default
incorporates the behavior being tested for.
* test/lisp/erc/erc-scenarios-base-buffer-display.el
(erc-scenarios-base-buffer-display--reconnect-common):
* test/lisp/erc/erc-scenarios-base-compat-rename-bouncer.el
(erc-scenarios-common--base-compat-no-rename-bouncer):
* test/lisp/erc/erc-scenarios-base-netid-bouncer-recon-base.el
(erc-scenarios-base-netid-bouncer--recon-base):
* test/lisp/erc/erc-scenarios-base-netid-bouncer-recon-both.el
(erc-scenarios-base-netid-bouncer--recon-both):
* test/lisp/erc/erc-scenarios-base-netid-bouncer-recon-id.el
(erc-scenarios-base-netid-bouncer--reconnect-id-foo)
(erc-scenarios-base-netid-bouncer--reconnect-id-bar):
* test/lisp/erc/erc-scenarios-base-reconnect.el
(erc-scenarios-base-reconnect-timer)
(erc-scenarios-base-cancel-reconnect):
* test/lisp/erc/erc-scenarios-services-misc.el
(erc-scenarios-services-misc--reconnect-retry-nick):
* test/lisp/erc/erc-scenarios-stamp.el
(erc-scenarios-stamp--date-mode/reconnect): Explicitly bind
`erc-server-reconnect-function' to `erc-server-delayed-reconnect', the
former default, which does not do any probing.  (Bug#62044)

lisp/erc/erc-backend.el

@@ -429,15 +429,16 @@ this value to 120 or greater and/or exploring the option
 means of handling this situation on some servers."
   :type 'number)
 
-(defcustom erc-server-reconnect-function 'erc-server-delayed-reconnect
+(defcustom erc-server-reconnect-function 'erc-server-prefer-check-reconnect
   "Function called by the reconnect timer to create a new connection.
 Called with a server buffer as its only argument.  Potential uses
 include exponential backoff and probing for connectivity prior to
 dialing.  Use `erc-schedule-reconnect' to instead try again later
 and optionally alter the attempts tally."
-  :package-version '(ERC . "5.5")
+  :package-version '(ERC . "5.6.1")
   :type '(choice (function-item erc-server-delayed-reconnect)
                  (function-item erc-server-delayed-check-reconnect)
+                 (function-item erc-server-prefer-check-reconnect)
                  function))
 
 (defcustom erc-split-line-length 440
@@ -879,7 +880,7 @@ Expect BUFFER to be the server buffer for the current connection."
              (sentinel (lambda (proc event)
                          (pcase event
                            ("open\n"
-                            (run-at-time nil nil #'send-string proc
+                            (run-at-time nil nil #'process-send-string proc
                                          (format "PING %d\r\n"
                                                  (time-convert nil 'integer))))
                            ((or "connection broken by remote peer\n"
@@ -901,6 +902,19 @@ Expect BUFFER to be the server buffer for the current connection."
               (set-process-sentinel proc sentinel))
           (file-error (funcall reschedule nil)))))))
 
+(defvar erc--server-delayed-check-connectors
+  '(erc-open-tls-stream erc-open-network-stream)
+  "Functions compatible with `erc-server-delayed-check-reconnect'.")
+
+(defun erc-server-prefer-check-reconnect (buffer)
+  "Defer to another reconnector based on BUFFER's `erc-session-connector'.
+Prefer `erc-server-delayed-check-reconnect' if the connector is known to
+be \"check-aware\".  Otherwise, use `erc-server-delayed-reconnect'."
+  (if (memq (buffer-local-value 'erc-session-connector buffer)
+            erc--server-delayed-check-connectors)
+      (erc-server-delayed-check-reconnect buffer)
+    (erc-server-delayed-reconnect buffer)))
+
 (defun erc-server-filter-function (process string)
   "The process filter for the ERC server."
   (with-current-buffer (process-buffer process)